% (find-LATEX "2026cwm.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2026cwm.tex" :end))
% (defun C () (interactive) (find-LATEXsh "lualatex 2026cwm.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2026cwm.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2026cwm.pdf"))
% (defun e () (interactive) (find-LATEX "2026cwm.tex"))
% (defun o () (interactive) (find-LATEX "2026cwm.tex"))
% (defun u () (interactive) (find-latex-upload-links "2026cwm"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun d0 () (interactive) (find-ebuffer "2026cwm.pdf"))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun oe () (interactive) (find-2a '(o) '(e)))
%          (code-eec-LATEX "2026cwm")
% (find-pdf-page   "~/LATEX/2026cwm.pdf")
% (find-sh0 "cp -v  ~/LATEX/2026cwm.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2026cwm.pdf /tmp/pen/")
%     (find-xournalpp "/tmp/2026cwm.pdf")
%   file:///home/edrx/LATEX/2026cwm.pdf
%               file:///tmp/2026cwm.pdf
%           file:///tmp/pen/2026cwm.pdf
%  http://anggtwu.net/LATEX/2026cwm.pdf
% https://anggtwu.net/LATEX/2026cwm.pdf
% (find-LATEX "2019.mk")
% (find-Deps1-links "Caepro5 Piecewise2 Maxima2")
% (find-Deps1-cps   "Caepro5 Piecewise2 Maxima2 DiagForth1")
% (find-Deps1-anggs "Caepro5 Piecewise2 Maxima2")
% (find-MM-aula-links "2026cwm" "2" "cwm2026" "cwm")

% «.geometry»			(to "geometry")
% «.edrx26a»			(to "edrx26a")
%  «.biber»			(to "biber")
% «.edrx26b»			(to "edrx26b")
% «.edrx26c»			(to "edrx26c")
% «.defs»			(to "defs")
% «.footer»			(to "footer")
% «.defs-T-and-B»		(to "defs-T-and-B")
%
% «.title»			(to "title")
% «.toc»			(to "toc")
% «.links»			(to "links")
% «.defs-adj-names»		(to "defs-adj-names")
% «.defs-adj-diag»		(to "defs-adj-diag")
% «.defs-adj-minipage»		(to "defs-adj-minipage")
% «.defs-abcells»		(to "defs-abcells")
% «.defs-adjs-conj-pure»	(to "defs-adjs-conj-pure")
% «.defs-adjs-conj-mixed»	(to "defs-adjs-conj-mixed")
% «.defs-adjs-conj-sigma-tau»	(to "defs-adjs-conj-sigma-tau")
% «.defs-conj-cell-eqs»		(to "defs-conj-cell-eqs")
% «.adjunctions»		(to "adjunctions")
% «.adjunctions-conj-pure»	(to "adjunctions-conj-pure")
% «.adjunctions-conj-mixed»	(to "adjunctions-conj-mixed")
%
% «.writetoc»			(to "writetoc")
% «.references»			(to "references")

% ;-- defs
\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
\usepackage{colorweb}                  % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-LATEX "dednat7-test1.tex")
%\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy         % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
% «geometry»  (to ".geometry")
% (find-es "tex" "geometry")
\usepackage[a6paper, landscape,
            top=1.5cm, bottom=.25cm, left=1cm, right=1cm, includefoot
           ]{geometry}
%
% «edrx26a»  (to ".edrx26a")
\usepackage{edrx26a}              % (find-LATEX "edrx26a.sty")
%
% «biber»  (to ".biber")
%\usepackage[backend=biber,
%   style=alphabetic]{biblatex}    % (find-es "tex" "biber")
%\addbibresource{catsem-ab.bib}    % (find-LATEX "catsem-ab.bib")
%\addbibresource{education.bib}    % (find-LATEX "education.bib")
%
\begin{document}

% «edrx26b»  (to ".edrx26b")
\input edrx26b.tex                % (find-LATEX "edrx26b.tex")

% «edrx26c»  (to ".edrx26c")
% (find-LATEX     "edrx26c.tex")
%L processsubfile "edrx26c.tex"   -- runs the "%L"s
\input             edrx26c         % loads the defs

% «defs»  (to ".defs")
% (find-LATEX "edrx21defs.tex" "colors")
% (find-LATEX "edrx21.sty")

% «footer»  (to ".footer")
% (find-LATEX "edrxheadfoot.tex")
\def\drafturl{http://anggtwu.net/LATEX/2026-1-C2.pdf}
\def\drafturl{http://anggtwu.net/2026.1-C2.html}
\def\draftfooter{\tiny \href{\drafturl}{\jobname{}} \ColorBrown{\shorttoday{} \hours}}

% «defs-T-and-B»  (to ".defs-T-and-B")
\long\def\ColorDarkOrange#1{{\color{orange!90!black}#1}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){\ColorRed{\bf(Total: #1 pts)}}
\def\B       (#1 pts){\ColorDarkOrange{\bf(#1 pts)}}


%L require "DiagForth1"  -- (find-angg "LUA/DiagForth1.lua")
\pu


% ;-- title
%  _____ _ _   _                               
% |_   _(_) |_| | ___   _ __   __ _  __ _  ___ 
%   | | | | __| |/ _ \ | '_ \ / _` |/ _` |/ _ \
%   | | | | |_| |  __/ | |_) | (_| | (_| |  __/
%   |_| |_|\__|_|\___| | .__/ \__,_|\__, |\___|
%                      |_|          |___/      
%
% «title»  (to ".title")
% (cwm2026p 1 "title")
% (cwm2026a   "title")

\thispagestyle{empty}

\begin{center}

\vspace*{1.2cm}

{\bf \Large Notes about Mac Lane's CWM}

\bsk

%Aula nn: ponha o título aqui
%
%\bsk

Eduardo Ochs - RCN/PURO/UFF

Psicopata do CEFET

\url{https://anggtwu.net/math-b.html}

\end{center}

%\newpage
% ;-- toc
% «toc»  (to ".toc")
% (to "writetoc")

% ;-- links
% «links»  (to ".links")
% (cwm2026p 2 "links")
% (cwm2026a   "links")

%{\bf Links}
%
%\scalebox{0.6}{\def\colwidth{16cm}\firstcol{
%}\anothercol{
%}}



% «defs-adj-names»  (to ".defs-adj-names")
\sa {adjunction: Edrx names} {
  \sa {A}     {A}
  \sa {B}     {B}
  \sa {L}     {L}
  \sa {R}     {R}
  \sa {LA}    {LA}
  \sa {RB}    {RB}
  \sa {catA}  {\catA}
  \sa {catB}  {\catB}
  \sa {sharp} {♯}
  \sa {flat}  {♭}
  }
\sa {adjunction: CWM names} {
  \sa {A}     {x}
  \sa {B}     {a}
  \sa {L}     {F}
  \sa {R}     {G}
  \sa {LA}    {Fx}
  \sa {RB}    {Ga}
  \sa {catA}  {X}
  \sa {catB}  {A}
  \sa {sharp} {φ}
  \sa {flat}  {ψ}
  }

% «defs-adj-diag»  (to ".defs-adj-diag")
%D diagram adjunction-1
%D 2Dx     100         +25
%D 2D  100 LA <======= A
%D 2D      |           |
%D 2D      |    <-->   |
%D 2D      v           v
%D 2D  +25 B =======> RB
%D 2D
%D 2D  +10 catB <=> catA
%D 2D
%D ren LA A ==> \ga{LA} \ga{A}
%D ren B RB ==> \ga{B} \ga{RB}
%D ren catB catA ==> \ga{catB} \ga{catA}
%D
%D (( LA A <-|
%D    LA B ->
%D    A RB ->
%D    B RB |->
%D
%D    LA RB harrownodes nil 20 nil <- sl^ .plabel= a \ga{flat}
%D    LA RB harrownodes nil 20 nil -> sl_ .plabel= b \ga{sharp}
%D
%D    catB catA <- sl^ .plabel= a \ga{L}
%D    catB catA -> sl_ .plabel= b \ga{R}
%D ))
%D enddiagram
\pu

\sa {adjunction diag Edrx} {{\ga{adjunction: Edrx names} \diag{adjunction-1}}}
\sa {adjunction diag CWM}  {{\ga{adjunction: CWM  names} \diag{adjunction-1}}}


% «defs-adj-minipage»  (to ".defs-adj-minipage")
%
\Sa {adjunction minipage} {{
  \begin{minipage}[t]{\colwidth}
    \par Let $\ga{catB}$ and $\ga{catA}$ be categories.
    \par An adjunction from $\ga{catB}$ to $\ga{catA}$
    \par is a triple $〈\ga{L},\ga{R},\ga{sharp}〉:\ga{catA} \rightharpoonup \ga{catB}$
    \par where $\ga{L}$ and $\ga{R}$ are functors
    %
    $$\ga{catA} \two/->`<-/^{\ga{L}}_{\ga{R}} \ga{catB}
    $$
    \par while $\ga{sharp}$ is a function
    %
    $$\ga{sharp}=\ga{sharp}_{\ga{A},\ga{B}}:
      \ga{catB}(\ga{LA},\ga{B}) ≅
      \ga{catA}(\ga{A},\ga{RB})
    $$
    \par which is natural in $\ga{A}$ and $\ga{B}$.
    \par Some authors write $\ga{L}⊣\ga{R}$.
  \end{minipage}
  }}
%
\sa {adjunction minipage Edrx} {{
  \def\colwidth{8cm}
  \ga{adjunction: Edrx names}
  \ga{adjunction minipage}
}}
\sa {adjunction minipage CWM} {{
  \def\colwidth{8cm}
  \ga{adjunction: CWM names}
  \ga{adjunction minipage}
}}



\def\Dn#1{{\scriptstyle \Downarrow \, #1}}

% «defs-abcells»  (to ".defs-abcells")
% (cwm2026a "defs-abcells")
% (find-angg "LUA/DiagForth1.lua" "relplace")
%L
%L forths["ab!"] = function ()
%L     node_a = ds:pick(1)
%L     node_b = ds:pick(0)
%L   end
%L forths["ab@"] = function ()
%L     ds:push(node_a)
%L     ds:push(node_b)
%L   end
%L
%L forths[".curve^^"] = function () forths[".curve="]("^32pt") end
%L forths[".curve^"]  = function () forths[".curve="]( "^8pt") end
%L forths[".curve_"]  = function () forths[".curve="]( "_8pt") end
%L forths[".curve__"] = function () forths[".curve="]("_32pt") end
%L
%L forthe["ab-name:"] = "e,e,w"
%L forths["ab-name:"] = function (dx,dy,TeX)
%L     forths["ab@"]()
%L     forths["midpoint"]()
%L     forths["relplace"](dx,dy,TeX)
%L   end
%L
%L y_eta = -10
%L y_eps =  10
%L
%L forthe["ab-name-up:"]   = "w"
%L forthe["ab-name-mid:"]  = "w"
%L forthe["ab-name-down:"] = "w"
%L forths["ab-name-up:"]   = function (tex) forths["ab-name:"](0,y_eta,tex) end
%L forths["ab-name-mid:"]  = function (tex) forths["ab-name:"](0,0,    tex) end
%L forths["ab-name-down:"] = function (tex) forths["ab-name:"](0,y_eps,tex) end
%L
%L forths["ab-up-eta"]    = function () dxyrun "ab-name-up:   \\Dn{η}" end
%L forths["ab-up-eta'"]   = function () dxyrun "ab-name-up:   \\Dn{η'}" end
%L forths["ab-down-eps"]  = function () dxyrun "ab-name-down: \\Dn{ε}" end
%L forths["ab-down-eps'"] = function () dxyrun "ab-name-down: \\Dn{ε'}" end
%L forths["ab-mid-id"]    = function () dxyrun "ab-name-mid:  \\Dn{\\id}" end
%L forths["ab-mid-sigma"] = function () dxyrun "ab-name-mid:  \\Dn{σ}" end
%L forths["ab-mid-tau"]   = function () dxyrun "ab-name-mid:  \\Dn{τ}" end
%L
%L forths["ab-mid-id-L"]  = function () dxyrun "ab-name-mid:  \\Dn{\\id_L}" end
%L forths["ab-mid-id-R"]  = function () dxyrun "ab-name-mid:  \\Dn{\\id_R}" end
%L forths["ab-mid-id-L'"] = function () dxyrun "ab-name-mid:  \\Dn{\\id_{L'}}" end
%L forths["ab-mid-id-R'"] = function () dxyrun "ab-name-mid:  \\Dn{\\id_{R'}}" end
%L
%L forths["ab-curve^^"] = function () dxyrun "ab@ -> .curve^^" end
%L forths["ab-curve__"] = function () dxyrun "ab@ -> .curve__" end
%L
%L forths["ab-eta"]  = function () dxyrun "ab-curve^^ ab-up-eta"    end
%L forths["ab-eta'"] = function () dxyrun "ab-curve^^ ab-up-eta'"   end
%L forths["ab-eps"]  = function () dxyrun "ab-curve__ ab-down-eps"  end
%L forths["ab-eps'"] = function () dxyrun "ab-curve__ ab-down-eps'" end
%L
%L forths["ab-sigma"] = function ()
%L     dxyrun "ab@ -> .curve^ .plabel= a L"
%L     dxyrun "ab@ => .curve_ .plabel= b L'"
%L     dxyrun "ab-name-mid: \\Dn{σ}"
%L   end
%L forths["ab-tau"] = function ()
%L     dxyrun "ab@ => .curve^ .plabel= a R'"
%L     dxyrun "ab@ -> .curve_ .plabel= b R"
%L     dxyrun "ab-name-mid: \\Dn{τ}"
%L   end


% «defs-adjs-conj-pure»  (to ".defs-adjs-conj-pure")
% The two adjunctions - the "pure" diagrams:
%
%         ___________                        ___________             
%        /           \                      /           \                 
%       /     ⇓ η     \                    /             \                
%      /               \    L             /               \    L  
%     A -----> B --R--> A -----> B   =   A -----> B  ⇓id   A -----> B
%         L     \               /            L     \               / 
%                \     ⇓ ε     /                    \             /  
%                 \___________/                      \___________/   
% 
%                  ___________                        ___________    
%                 /           \                      /           \   
%                /     ⇓ η     \                    /             \  
%         R     /               \            R     /               \ 
%     B -----> A --L--> B -----> A   =   B -----> A   ⇓id  B -----> A
%      \               /    R             \               /    R     
%       \     ⇓ ε     /                    \             /           
%        \___________/                      \___________/            
% 
% 
% 
%         ___________                        ___________             
%        /           \                      /           \                 
%       /     ⇓ η'    \                    /             \                
%      /               \    L'            /               \    L' 
%     A =====> B ==R'=> A =====> B   =   A =====> B  ⇓id   A =====> B
%         L'    \               /            L'    \               / 
%                \     ⇓ ε'    /                    \             /  
%                 \___________/                      \___________/   
% 
%                  ___________                        ___________    
%                 /           \                      /           \   
%                /     ⇓ η     \                    /             \  
%         R'    /               \            R'    /               \ 
%     B =====> A ==L'=> B ======> A  =   B =====> A   ⇓id  B ======> A
%      \               /    R'            \               /    R'     
%       \     ⇓ ε     /                    \             /           
%        \___________/                      \___________/            


%D diagram cells-pure-nw-se-1
%D 2Dx     100      +25      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /     ⇓ η     \           
%D 2D       /               \    L     
%D 2D  100 A0 ----> B1 -R--> A2 ----> B3
%D 2D          L     \               / 
%D 2D                 \     ⇓ ε     /  
%D 2D                  \___________/   
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 -> .plabel= b L
%D    B1 A2 -> .plabel= m R
%D    A2 B3 -> .plabel= a L
%D    A0 A2 ab!   ab-eta
%D    B1 B3 ab!   ab-eps
%D ))
%D enddiagram
\pu

%D diagram cells-pure'-nw-se-1
%D 2Dx     100      +25      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /     ⇓ η'    \           
%D 2D       /               \    L'    
%D 2D  100 A0 ====> B1 =R'=> A2 ====> B3
%D 2D          L'    \               / 
%D 2D                 \     ⇓ ε'    /  
%D 2D                  \___________/   
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L
%D    B1 A2 => .plabel= m R
%D    A2 B3 => .plabel= a L
%D    A0 A2 ab!   ab-eta'
%D    B1 B3 ab!   ab-eps'
%D ))
%D enddiagram
\pu

%D diagram cells-pure-nw-se-2
%D 2Dx     100      +25      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /             \           
%D 2D       /               \    L     
%D 2D  100 A0 ----> B1 ⇓ id  A2 ----> B3
%D 2D          L     \               / 
%D 2D                 \             /  
%D 2D                  \___________/   
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 -> .plabel= b L
%D    A2 B3 -> .plabel= a L
%D    A0 A2 ab!  ab-curve^^
%D    B1 A2 ab!  ab-mid-id-L
%D    B1 B3 ab!  ab-curve__
%D ))
%D enddiagram
\pu

%D diagram cells-pure'-nw-se-2
%D 2Dx     100      +25      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /             \           
%D 2D       /               \    L'     
%D 2D  100 A0 ====> B1 ⇓ id  A2 ====> B3
%D 2D          L'    \               / 
%D 2D                 \             /  
%D 2D                  \___________/   
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L'
%D    A2 B3 => .plabel= a L'
%D    A0 A2 ab!  ab-curve^^
%D    B1 A2 ab!  ab-mid-id-L'
%D    B1 B3 ab!  ab-curve__
%D ))
%D enddiagram
\pu

%D diagram cells-pure-ne-sw-1
%D 2Dx     100      +25      +25      +25
%D 2D  100               __________
%D 2D                   /          \
%D 2D                  /     ⇓ η    \
%D 2D          R      /              \ 
%D 2D  100 B0 ----> A1 -L--> B2 ----> A3
%D 2D       \               /     R
%D 2D        \     ⇓ ε     /
%D 2D         \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 -> .plabel= a R
%D    A1 B2 -> .plabel= m L
%D    B2 A3 -> .plabel= b R
%D    B0 B2 ab!  ab-eps
%D    A1 A3 ab!  ab-eta
%D ))
%D enddiagram
\pu

%D diagram cells-pure'-ne-sw-1
%D 2Dx     100      +25      +25      +25
%D 2D  100               __________
%D 2D                   /          \
%D 2D                  /     ⇓ η'   \
%D 2D          R'     /              \ 
%D 2D  100 B0 ====> A1 =L==> B2 ====> A3
%D 2D       \               /     R
%D 2D        \     ⇓ ε'    /
%D 2D         \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D    A1 B2 => .plabel= m L'
%D    B2 A3 => .plabel= b R'
%D    B0 B2 ab!  ab-eps'
%D    A1 A3 ab!  ab-eta'
%D ))
%D enddiagram
\pu

%D diagram cells-pure-ne-sw-2
%D 2Dx     100      +25      +25      +25
%D 2D  100               __________
%D 2D                   /          \
%D 2D                  /            \
%D 2D          R      /              \ 
%D 2D  100 B0 ----> A1  ⇓id  B2 ----> A3
%D 2D       \               /     R
%D 2D        \             /
%D 2D         \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 -> .plabel= a R
%D    B2 A3 -> .plabel= b R
%D    A1 A3 ab!  ab-curve^^
%D    A1 B2 ab!  ab-mid-id-R
%D    B0 B2 ab!  ab-curve__
%D ))
%D enddiagram
\pu

%D diagram cells-pure'-ne-sw-2
%D 2Dx     100      +25      +25      +25
%D 2D  100               __________
%D 2D                   /          \
%D 2D                  /            \
%D 2D          R'     /              \ 
%D 2D  100 B0 ====> A1  ⇓id  B2 ====> A3
%D 2D       \               /     R
%D 2D        \             /
%D 2D         \___________/
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D    B2 A3 => .plabel= b R'
%D    A1 A3 ab!  ab-curve^^
%D    A1 B2 ab!  ab-mid-id-R'
%D    B0 B2 ab!  ab-curve__
%D ))
%D enddiagram
\pu


% «defs-adjs-conj-mixed»  (to ".defs-adjs-conj-mixed")
% The two adjunctions - the "mixed" diagrams:
%
%         ___________                        ___________             
%        /           \                      /           \                 
%       /     ⇓ η'    \                    /             \                
%      /         =R'=> \    L             /               \    L  
%     A =====> B  ⇓ τ   A -----> B   =   A =====> B  ⇓ σ    A -----> B
%         L'    \ -R-->         /            L'    \               / 
%                \     ⇓ ε     /                    \             /  
%                 \___________/                      \___________/   
% 
%                  ___________                        ___________    
%                 /           \                      /           \   
%                /     ⇓ η     \                    /             \  
%         R'    / --L-->        \            R'    /               \ 
%     B =====> A  ⇓ σ   B -----> A   =   B =====> A   ⇓ τ  B -----> A
%      \         =L'=> /    R             \               /    R     
%       \     ⇓ ε'    /                    \             /           
%        \___________/                      \___________/            
% 
%
%         ___________                        ___________             
%        /           \                      /           \                 
%       /     ⇓ η'    \                    /      ⇓ η    \                
%      /         =R'=> \                  / --L-->        \
%     A =====> B  ⇓ τ   A            =   A   ⇓ σ  B --R--> A
%         L'     --R-->                    ==L'=>
%
%
%         R'     --L-->                    ==R'=>
%     B =====> A  ⇓ σ   B            =   B   ⇓ τ  A -----> B
%      \         =L'=> /                  \ --R-->        /
%       \     ⇓ ε'    /                    \     ⇓ ε     /           
%        \___________/                      \___________/            
% 
%
%D diagram cells-mixed-nw-se-before
%D 2Dx     100      +25      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /             \           
%D 2D       /         =R'=> \    L      
%D 2D  100 A0 ====> B1  ⇓ σ  A2 ----> B3
%D 2D          L'    \ --R-->        / 
%D 2D                 \             /  
%D 2D                  \___________/   
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L'
%D    A2 B3 -> .plabel= a L
%D    A0 A2 ab!  ab-eta'
%D    B1 A2 ab!  ab-tau
%D    B1 B3 ab!  ab-eps
%D ))
%D enddiagram
\pu

%D diagram cells-mixed-nw-se-after
%D 2Dx     100      +25      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /             \           
%D 2D       /               \    L      
%D 2D  100 A0 ====> B1  ⇓ σ  A2 ----> B3
%D 2D          L'    \               / 
%D 2D                 \             /  
%D 2D                  \___________/   
%D 2D
%D ren A0 B1 A2 B3 ==> \catA \catB \catA \catB
%D
%D (( A0 B1 => .plabel= b L'
%D    A2 B3 -> .plabel= a L
%D    A0 A2 ab!  ab-curve^^
%D    B1 A2 ab!  ab-name-mid: \Dn{σ}
%D    B1 B3 ab!  ab-curve__
%D ))
%D enddiagram
\pu

%D diagram cells-mixed-ne-sw-before
%D 2Dx     100      +25       +25      +25
%D 2D  100               ___________
%D 2D                   /           \
%D 2D                  /      ⇓ η    \
%D 2D           R'    / -L-->         \
%D 2D  100 B0 ====> A1  ⇓ σ   B2 --R-> A3
%D 2D        \        ==L'=> / 
%D 2D         \             /  
%D 2D          \___________/   
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D    B2 A3 -> .plabel= b R
%D    A1 A3 ab!  ab-eta
%D    A1 B2 ab!  ab-sigma
%D    B0 B2 ab!  ab-eps'
%D ))
%D enddiagram
\pu

%D diagram cells-mixed-ne-sw-after
%D 2Dx     100      +25       +25      +25
%D 2D  100               ___________
%D 2D                   /           \
%D 2D                  /      ⇓ η    \
%D 2D           R'    / -L-->         \
%D 2D  100 B0 ====> A1  ⇓ σ   B2 --R-> A3
%D 2D        \        ==L'=> / 
%D 2D         \             /  
%D 2D          \___________/   
%D 2D
%D ren B0 A1 B2 A3 ==> \catB \catA \catB \catA
%D
%D (( B0 A1 => .plabel= a R'
%D    B2 A3 -> .plabel= b R
%D    A1 A3 ab!  ab-curve^^
%D    A1 B2 ab!  ab-name-mid: \Dn{τ}
%D    B0 B2 ab!  ab-curve__
%D ))
%D enddiagram
\pu

%D diagram cells-flip-eta-before
%D 2Dx     100      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /   ⇓ η'      \           
%D 2D       /         =R'=> \
%D 2D  100 A0 ====> B1  ⇓ σ  A2
%D 2D          L'     --R-->
%D 2D
%D ren A0 B1 A2 ==> \catA \catB \catA
%D
%D (( A0 B1 => .plabel= b L'
%D    A0 A2 ab!  ab-eta'
%D    B1 A2 ab!  ab-tau
%D ))
%D enddiagram
\pu

%D diagram cells-flip-eta-after
%D 2Dx     100      +25      +25
%D 2D  100     ___________             
%D 2D         /           \            
%D 2D        /      ⇓ η    \           
%D 2D       / -L-->         \
%D 2D  100 A0  ⇓ σ  B1 --R-> A2
%D 2D         =L'->   
%D 2D
%D ren A0 B1 A2 ==> \catA \catB \catA
%D
%D (( A0 B1 ab!  ab-sigma
%D    A0 A2 ab!  ab-eta
%D    B1 A2 => .plabel= b R
%D ))
%D enddiagram
\pu



%D diagram cells-flip-eps-before
%D 2Dx     100      +25       +25
%D 2D
%D 2D           R'     --L-->
%D 2D  100 B0 ====> A1  ⇓ σ   B2
%D 2D        \        ==L'=> /
%D 2D         \   ⇓ ε'      /  
%D 2D          \___________/   
%D 2D
%D ren B0 A1 B2 ==> \catB \catA \catB
%D
%D (( B0 A1 => .plabel= a R'
%D    A1 B2 ab!  ab-sigma
%D    B0 B2 ab!  ab-eps'
%D ))
%D enddiagram
\pu

%D diagram cells-flip-eps-after
%D 2Dx     100      +25       +25
%D 2D
%D 2D         ==R'=>     L
%D 2D  100 B0  ⇓ τ  A1 -----> B2
%D 2D        \ --R->         / 
%D 2D         \     ⇓ ε     /  
%D 2D          \___________/   
%D 2D
%D ren B0 A1 B2 ==> \catB \catA \catB
%D
%D (( A1 B2 -> .plabel= a L
%D    B0 A1 ab!  ab-tau
%D    B0 B2 ab!  ab-eps
%D ))
%D enddiagram
\pu


% «defs-adjs-conj-sigma-tau»  (to ".defs-adjs-conj-sigma-tau")
% The cell diagrams for σ and τ:
%
%D diagram cells-sigma
%D 2Dx     100       +25
%D 2D         --L-->   
%D 2D  100 A0  ⇓ σ   B1
%D 2D         ==L'=>  
%D 2D
%D ren A0 B1 ==> \catA \catB
%D
%D (( A0 B1 ab!  ab-sigma
%D ))
%D enddiagram
\pu

%D diagram cells-tau
%D 2Dx     100       +25
%D 2D         ==R'==>   
%D 2D  100 B0  ⇓ τ   A1
%D 2D         ==L'=>  
%D 2D
%D ren B0 A1 ==> \catB \catA
%D
%D (( B0 A1 ab!  ab-tau
%D ))
%D enddiagram
\pu

% «defs-conj-cell-eqs»  (to ".defs-conj-cell-eqs")
%
\sa {Conjungate adjunctions: cell equations (pure)} {
  \begin{array}{rcl}
    \diag{cells-pure-nw-se-1} &=&
    \diag{cells-pure-nw-se-2} \\
    \diag{cells-pure-ne-sw-1} &=&
    \diag{cells-pure-ne-sw-2} \\\\
    \diag{cells-pure'-nw-se-1} &=&
    \diag{cells-pure'-nw-se-2} \\
    \diag{cells-pure'-ne-sw-1} &=&
    \diag{cells-pure'-ne-sw-2} \\
  \end{array}
  }
\sa {Conjungate adjunctions: cell equations (mixed)} {
  \begin{array}{rcl}
    \diag{cells-mixed-nw-se-before} &=&
    \diag{cells-mixed-nw-se-after} \\
    \diag{cells-mixed-ne-sw-before} &=&
    \diag{cells-mixed-ne-sw-after} \\
    \diag{cells-flip-eta-before} &=&
    \diag{cells-flip-eta-after} \\
    \diag{cells-flip-eps-before} &=&
    \diag{cells-flip-eps-after} \\
  \end{array}
  }


\newpage
% ;-- adjunctions
% «adjunctions»  (to ".adjunctions")
% (cwm2026p 2 "adjunctions")
% (cwm2026a    "adjunctions")
% (find-books "__cats/__cats.el" "maclane" "80" "Definition. Let A and X be categories.")
% (find-symbolspage 70 "Table 139: Harpoons")
% (find-symbolstext 70 "Table 139: Harpoons")

\SLIDE{adjunctions}

\scalebox{0.4}{\def\colwidth{18cm}\firstcol{

$$
\begin{array}{ccc}
  \ga{adjunction minipage CWM} &&
  \ga{adjunction minipage Edrx} \\
  \\
  \\
  \scalebox{2.0}{$\ga{adjunction diag CWM}$} &&
  \scalebox{2.0}{$\ga{adjunction diag Edrx}$} \\
\end{array}
$$

}\anothercol{
}}



\newpage
% ;-- adjunctions-conj-pure
% «adjunctions-conj-pure»  (to ".adjunctions-conj-pure")
% (cwm2026p 3 "adjunctions-conj-pure")
% (cwm2026a   "adjunctions-conj-pure")
% (misp 51 "2-category-of-cats")
% (misa    "2-category-of-cats")

\SLIDE{Conjugate adjunctions: cell equations (``pure'')}

\scalebox{0.6}{\def\colwidth{9.5cm}\firstcol{

$$\ga{Conjungate adjunctions: cell equations (pure)}
$$

}\anothercol{

\vspace*{2cm}

$$\begin{array}{rcl}
  \bmat{η,L\\L,ε} &=& [L] \\\\
  \bmat{R,η\\ε,R} &=& [R] \\\\
  \bmat{η',L'\\L',ε'} &=& [L'] \\\\
  \bmat{R',η'\\ε',R'} &=& [R'] \\\\
  \end{array}
$$

}}


\newpage
% ;-- adjunctions-conj-mixed
% «adjunctions-conj-mixed»  (to ".adjunctions-conj-mixed")
% (cwm2026p 4 "adjunctions-conj-mixed")
% (cwm2026a   "adjunctions-conj-mixed")

\SLIDE{Conjugate adjunctions: cell equations (``mixed'')}

\scalebox{0.6}{\def\colwidth{9.5cm}\firstcol{

$$\ga{Conjungate adjunctions: cell equations (mixed)}
$$

}\anothercol{

\vspace*{2cm}

$$\begin{array}{rcl}
  \bmat{η',L \\ L',τ,L \\ L',ε} &=& [σ] \\\\
  \bmat{R',η \\ R',σ,R \\  ε',R} &=& [τ] \\\\
  \bmat{η' \\ L',τ} &=& \bmat{η \\ σ,R} \\\\
  \bmat{R',σ \\ ε'} &=& \bmat{τ,L \\ ε} \\\\
  \end{array}
$$

}}



% (misp 51 "2-category-of-cats")
% (misa    "2-category-of-cats")
% (find-riehlccpage (+ 18  44) "1.7. The 2-category of categories")


\newpage

% (misp 24 "internal-view-NT")
% (misa    "internal-view-NT")

%D diagram generic-NT
%D 2Dx     100 +25
%D 2D  100     FA
%D 2D  +10 A
%D 2D  +10     GA
%D 2D
%D ren A FA GA ==> \ga{A} \ga{FA} \ga{GA}
%D
%D (( A FA |->
%D    A GA |->
%D    FA GA -> .plabel= r \ga{TA}
%D    A FA GA midpoint |->
%D ))
%D enddiagram
\pu

\sa {sigma NT} {{
  \sa {A} {A}
  \sa {FA} {LA}
  \sa {GA} {L'A}
  \sa {TA} {σ_A}
  \diag{generic-NT}
}}
\sa {tau NT} {{
  \sa {A} {B}
  \sa {FA} {R'B}
  \sa {GA} {RB}
  \sa {TA} {τ_B}
  \diag{generic-NT}
}}

%D diagram sigma-triangle
%D 2Dx     100 +25
%D 2D  100 LA  L'A
%D 2D
%D 2D  +20     B
%D 2D
%D # ren ==>
%D
%D (( LA L'A -> .plabel= a σ_A
%D    L'A  B -> .plabel= r f
%D    LA   B -> .plabel= l f∘σ_A
%D ))
%D enddiagram
\pu

%D diagram tau-triangle
%D 2Dx     100 +25
%D 2D  100 A
%D 2D
%D 2D  +20 R'B RB
%D 2D
%D # ren ==>
%D
%D (( A  R'B -> .plabel= l g
%D    R'B RB -> .plabel= b τ_B
%D    A   RB -> .plabel= r τ_B∘g
%D ))
%D enddiagram
\pu

\def\AoverB #1#2{\pmat{#1\\↓\\#2}}
\sa {L'A/B} {\AoverB{L'A}{B}}
\sa  {LA/B} {\AoverB {LA}{B}}
\sa {A/R'B} {\AoverB{A}{R'B}}
\sa {A/RB}  {\AoverB{A} {RB}}

%D diagram cwm-p100-6-edrx-1
%D 2Dx     100     +40
%D 2D  100 A0 <--- A1
%D 2D      |        |
%D 2D      v        v
%D 2D  +40 A2 ---> A3
%D 2D
%D ren A0 A1 ==> \AoverB{L'A}{B} \AoverB{A}{R'B}
%D ren A2 A3 ==> \AoverB{LA}{B}  \AoverB{A}{RB}
%D
%D (( A0 A1 <- .plabel= a ♭' sl^
%D    A0 A1 -> .plabel= b ♯' sl_
%D    A0 A2 -> .plabel= l ∘σ_A
%D    A1 A3 -> .plabel= r τ_B∘
%D    A2 A3 <- .plabel= a ♭ sl^
%D    A2 A3 -> .plabel= b ♯ sl_
%D ))
%D enddiagram
\pu

%D diagram cwm-p100-6-edrx-2
%D 2Dx     100     +40
%D 2D  100 A0 <--- A1
%D 2D      |        |
%D 2D      v        v
%D 2D  +40 A2 ---> A3
%D 2D
%D ren A0 A1 ==> \AoverB{L'R'B}{B} \AoverB{R'B}{R'B}
%D ren A2 A3 ==> \AoverB{LR'B}{B}  \AoverB{R'B}{RB}
%D
%D (( A0 A1 <- .plabel= a ♭'
%D    A0 A2 -> .plabel= l ∘σ_{R'B}
%D    A1 A3 -> .plabel= r τ_B∘
%D    A2 A3 -> .plabel= b ♯
%D ))
%D enddiagram
\pu

%D diagram cwm-p100-6-edrx-2-b
%D 2Dx     100     +80
%D 2D  100 A0 <--- A1
%D 2D      |        |
%D 2D  +25 v       A3'
%D 2D  +10 A2 ---> A3
%D 2D
%D ren A0 A1  ==> ε'_B \id_{R'B}
%D ren    A3' ==> τ_B=[τ]B
%D ren A2 A3  ==> \ga{cwm-p100-6-edrx-2-b-SW} \ga{cwm-p100-6-edrx-2-b-SE}
%D
%D (( A0 A1  <-| .plabel= a ♭'
%D    A0 A2  |-> .plabel= l ∘σ_{R'B}
%D    A1 A3' |-> .plabel= r τ_{B}∘
%D    A2 A3  |-> .plabel= b ♯
%D ))
%D enddiagram
\pu

\sa {cwm-p100-6-edrx-2-b-SW} {{
  \begin{array}[t]{l}
    ε'_B∘σ_{R'B} \\
    = [ε']B∘[R',σ]B \\
    = \bsm{R',σ \\ ε'}B \\
  \end{array}
}}

\sa {cwm-p100-6-edrx-2-b-SE} {{
  \begin{array}[t]{l}
    R(\bsm{R',σ \\ ε'}B) ∘ η_{RB} \\
    = [\bsm{R',σ \\ ε'},R]B ∘ [R,η]B \\
    = \bsm{R',σ,R \\ ε',R}B ∘ [R,η]B \\
    = \bsm{R,η \\ R',σ,R \\ ε',R}B \\
  \end{array}
}}


\scalebox{0.5}{\def\colwidth{9cm}\firstcol{

$$\begin{array}{cc}
  \diag{cells-sigma} & \ga {sigma NT} \\
  \diag{cells-tau}   & \ga {tau   NT} \\
  \end{array}
$$

$$\diag{sigma-triangle}
  \quad
  \diag{cwm-p100-6-edrx-1}
  \quad
  \diag{tau-triangle}
$$

$$\diag{cwm-p100-6-edrx-2} \qquad
  \diag{cwm-p100-6-edrx-2-b}
$$

}\anothercol{
}}









$$\pu
$$













% sigma and tau:
% (find-books "__cats/__cats.el" "maclane" "99" "7. Transformations of Adjoints")
% (find-books "__cats/__cats.el" "maclane" "100" "are said to be conjugate (for the given adjunctions) when the diagram")





% (find-es "diagxy" "diagxyto")
% (find-es "xypic" "two-and-three")
% (find-diagxypage 27 "\\two/<-< `<-< /^f_g")
% (find-diagxytext 27 "\\two/<-< `<-< /^f_g")








% ;-- writetoc
% «writetoc»  (to ".writetoc")
\directlua{toclines:writetoc()}
% Writes in: (find-LATEXfile "2026cwm.mytoc")
% See: (to "toc")

% ;-- references
% «references»  (to ".references")
%\printbibliography

% ;-- write-dnt-file
% «write-dnt-file»  (to ".write-dnt-file")
% (find-fline "~/LATEX/" "2026cwm.dnt")
% (find-fline    "~/LATEX/2026cwm.dnt")
%L write_dnt_file       ("2026cwm.dnt")
\pu


\GenericWarning{Success:}{Success!!!}  % Used by `M-x cv'

\end{document}

% (find-pdfpages2-links "~/LATEX/" "2026cwm")


% Local Variables:
% coding: utf-8-unix
% outline-regexp: "% +;--"
% ee-tla: "cwm"
% ee-tla: "cwm2026"
% End:
