% (find-LATEX "2023ends-and-coends.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2023ends-and-coends.tex" :end))
% (defun C () (interactive) (find-LATEXSH "lualatex 2023ends-and-coends.tex" "Success!!!"))
% (defun D () (interactive) (find-pdf-page      "~/LATEX/2023ends-and-coends.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2023ends-and-coends.pdf"))
% (defun e () (interactive) (find-LATEX "2023ends-and-coends.tex"))
% (defun u () (interactive) (find-latex-upload-links "2023ends-and-coends"))
% (defun v () (interactive) (find-2a '(e) '(d)))
% (defun cv () (interactive) (C) (ee-kill-this-buffer) (v) (g))
% (defun d0 () (interactive) (find-ebuffer "2023ends-and-coends.pdf"))
%          (code-eec-LATEX "2023ends-and-coends")
% (find-pdf-page   "~/LATEX/2023ends-and-coends.pdf")
% (find-sh0 "cp -v  ~/LATEX/2023ends-and-coends.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2023ends-and-coends.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2023ends-and-coends.pdf
%               file:///tmp/2023ends-and-coends.pdf
%           file:///tmp/pen/2023ends-and-coends.pdf
%  http://anggtwu.net/LATEX/2023ends-and-coends.pdf
% (find-LATEX "2019.mk")
% (find-lualatex-links "2023ends-and-coends" "ends")

\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-dn6 "preamble6.lua" "preamble0")
\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy        % For 2D diagrams ("%D" lines)
\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx21}               % (find-LATEX "edrx21.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrx21chars.tex            % (find-LATEX "edrx21chars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%\input 2017planar-has-defs.tex    % (find-LATEX "2017planar-has-defs.tex")
%
%\usepackage[backend=biber,
%   style=alphabetic]{biblatex}            % (find-es "tex" "biber")
%\addbibresource{catsem-slides.bib}        % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\begin{document}

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

% %L dofile "edrxtikz.lua"  -- (find-LATEX "edrxtikz.lua")
% %L dofile "edrxpict.lua"  -- (find-LATEX "edrxpict.lua")
% \pu

%\printbibliography


%D diagram ??
%D 2Dx     100  +20  +20  +20  +20    +20  +20  +20  +20  +20
%D 2D  100      A1 -----_                  D1 -----_              
%D 2D           |  \      \                   \      \       
%D 2D  +15      |    A2 - A3                    D2 - D3      
%D 2D           |    |     |                                 
%D 2D  +10      A4 --|---_ |               D4 ------_        
%D 2D              \ |    \|                  \      \       
%D 2D  +15           A5 - A6                    D5 - D6      
%D 2D                                                     
%D 2D                                                     
%D 2D  +20           B1 - B5                    E1 - E5      
%D 2D              /    X    \                /         \    
%D 2D  +20 C1   B0 - B4   B3 - B7     F1   E0             E7 
%D 2D         \    \    X    /           \    \         /    
%D 2D  +20      C2   B2 - B6               F2   E2 - E6      
%D # ren ==>
%D
%D (( A1 A2 |-> A1 A3 |-> A2 A3 ->
%D    A4 A5 |-> A4 A6 |-> A5 A6 ->
%D    A1 A4  -> A2 A5  -> A3 A6 ->
%D    
%D    D1 D2 |-> D1 D3 |-> D2 D3 ->
%D    D4 D5 |-> D4 D6 |-> D5 D6 ->
%D    D1 D4  -> D2 D5  -> D3 D6 ->
%D
%D    B0 B1 -> B2 B3 -> B4 B5 -> B6 B7 ->
%D    B0 B2 -> B1 B3 -> B4 B6 -> B5 B7 ->
%D    B0 B4 -> B1 B5 -> B2 B6 -> B3 B7 ->
%D
%D    C1 C2 ->
%D
%D    E0 E1 -> E1 E5 -> E5 E7 ->
%D    E0 E2 -> E2 E6 -> E6 E7 ->
%D
%D    F1 F2 ->
%D ))
%D enddiagram
%D
%$$\pu
%  \diag{??}
%$$

\def\tntoo{\ton{\bullet\bullet}}


$F,G: \catC^\op×\catC → \catD$

$T: F \tnto G$

$A,B,C,D ∈ \catC$

$f:A→B$

$g:C→A$

$h:B→D$

$(A,B),(C,D)∈\catC^\op×\catC$

$(f,g):(A,B)→(C,D)$

\msk

$U: F \tntoo G$

$UA:FAA→GAA$

$UB:FBB→GBB$


%D diagram ???
%D 2Dx     100  +25  +25  +25  +25    +25  +25  +25  +25  +25
%D 2D  100      A1 -----_                  D1 -----_              
%D 2D           |  \      \                   \      \       
%D 2D  +15      |    A2 - A3                    D2 - D3      
%D 2D           |    |     |                                 
%D 2D  +10      A4 --|---_ |               D4 ------_        
%D 2D              \ |    \|                  \      \       
%D 2D  +15           A5 - A6                    D5 - D6      
%D 2D                                                     
%D 2D                                                     
%D 2D  +20           B1 - B5                    E1 - E5      
%D 2D              /    X    \                /         \    
%D 2D  +20 C1   B0 - B4   B3 - B7     F1   E0             E7 
%D 2D         \    \    X    /           \    \         /    
%D 2D  +20      C2   B2 - B6               F2   E2 - E6      
%D 2D
%D ren A1 A2 A3    ==> (A,B) FAB GAB
%D ren A4 A5 A6    ==> (C,D) FCD GCD
%D ren B0 B1 B2 B3 ==> FBA FAA FBB FAB
%D ren B4 B5 B6 B7 ==> GBA GAA GBB GAB
%D ren C1 C2       ==> A B
%D ren D1 D2 D3    ==> (A,A) FAA GAA
%D ren D4 D5 D6    ==> (B,B) FBB GBB
%D ren E0 E1 E2    ==> FBA FAA FBB
%D ren    E5 E6 E7 ==>     GAA GBB GAB
%D ren F1 F2       ==> A B
%D
%D (( A1 A2 |-> A1 A3 |-> A2 A3 ->
%D    A4 A5 |-> A4 A6 |-> A5 A6 ->
%D    A1 A4  -> A2 A5  -> A3 A6 ->
%D    
%D    D1 D2 |-> D1 D3 |-> D2 D3 ->
%D    D4 D5 |-> D4 D6 |-> D5 D6 ->
%D
%D    B0 B1 -> B2 B3 -> B4 B5 -> B6 B7 ->
%D    B0 B2 -> B1 B3 -> B4 B6 -> B5 B7 ->
%D    B0 B4 -> B1 B5 -> B2 B6 -> B3 B7 ->
%D
%D    C1 C2 ->
%D
%D    E0 E1 -> E1 E5 -> E5 E7 ->
%D    E0 E2 -> E2 E6 -> E6 E7 ->
%D
%D    F1 F2 ->
%D ))
%D enddiagram
%D
$$\pu
  \diag{???}
$$



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

\end{document}


% Local Variables:
% coding: utf-8-unix
% ee-tla: "ends"
% End:
