% (find-LATEX "2020seelyhyp-poster.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020seelyhyp-poster.tex" :end))
% (defun d () (interactive) (find-pdf-page      "~/LATEX/2020seelyhyp-poster.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020seelyhyp-poster.pdf"))
% (defun e () (interactive) (find-LATEX "2020seelyhyp-poster.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020seelyhyp-poster"))
% (defun v () (interactive) (find-2a '(e) '(d)) (g))
% (find-pdf-page   "~/LATEX/2020seelyhyp-poster.pdf")
% (find-sh0 "cp -v  ~/LATEX/2020seelyhyp-poster.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2020seelyhyp-poster.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2020seelyhyp-poster.pdf
%               file:///tmp/2020seelyhyp-poster.pdf
%           file:///tmp/pen/2020seelyhyp-poster.pdf
% http://angg.twu.net/LATEX/2020seelyhyp-poster.pdf
% (find-LATEX "2019.mk")

% «.adjoints-generic»	(to "adjoints-generic")
% «.adjoints-quants»	(to "adjoints-quants")
% «.adjoints-equal»	(to "adjoints-equal")
% «.adjoints-f»		(to "adjoints-f")

\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{edrx15}               % (find-LATEX "edrx15.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
%\usepackage[backend=biber,
%   style=alphabetic]{biblatex}            % (find-es "tex" "biber")
%\addbibresource{catsem-slides.bib}        % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "limp-abx")
\DeclareFontFamily{U}{matha}{\hyphenchar\font45}
\DeclareFontShape{U}{matha}{m}{n}{
      <5> <6> <7> <8> <9> <10> gen * matha
      <10.95> matha10 <12> <14.4> <17.28> <20.74> <24.88> matha12
      }{}
\DeclareSymbolFont{matha}{U}{matha}{m}{n}
\DeclareMathSymbol{\varsubset}{3}{matha}{"80}
\DeclareMathSymbol{\varsupset}{3}{matha}{"81}
\def\limp{\varsupset}
\catcode`⊸=13 \def⊸{\limp}
%
% (find-es "tex" "geometry")
\begin{document}

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


\def\pded#1{\left(\cded{#1}\right)}
\def\pdedscale#1#2{\scalebox{#1}{$\pded{#2}$}}
\def\eo  #1#2{#1{=}#2}
\def\ea#1#2#3{#1{=}#2∧#3}
\def\ei#1#2#3{#1{=}#2⊸#3}



%     _       _  _    __                              _      
%    / \   __| |(_)  / _|   __ _  ___ _ __   ___ _ __(_) ___ 
%   / _ \ / _` || | | |_   / _` |/ _ \ '_ \ / _ \ '__| |/ __|
%  / ___ \ (_| || | |  _| | (_| |  __/ | | |  __/ |  | | (__ 
% /_/   \_\__,_|/ | |_|    \__, |\___|_| |_|\___|_|  |_|\___|
%             |__/         |___/                             
%
% «adjoints-generic»  (to ".adjoints-generic")

Adjoints to an arbitrary $f^*$:

\bsk

%D diagram adjs-generic
%D 2Dx     100    +40
%D 2D  100 A0 |-> A1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 A2 |-> A3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 B0 <-| B1
%D 2D      |       |
%D 2D      |  <-|  |
%D 2D      |       |
%D 2D  +25 B2 <-| B3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 C0 |-> C1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 C2 |-> C3
%D 2D
%D 2D  +20 D0 <=> D1
%D 2D
%D 2D  +20 E2 --> E3
%D 2D
%D ren A0 A1 ==> A_1 Σ_fA_1
%D ren A2 A3 ==> A_2 Σ_fA_2
%D ren B0 B1 ==> f^*B_1 B_1
%D ren B2 B3 ==> f^*B_2 B_2
%D ren C0 C1 ==> C_1 Π_fC_1
%D ren C2 C3 ==> C_2 Π_fC_2
%D ren D0 D1 ==> 𝐛P(X) 𝐛P(Y)
%D ren E2 E3 ==> X Y
%D
%D (( A0 A1 |->
%D    A2 A3 |->
%D    B0 B1 <-|
%D    B2 B3 <-|
%D    C0 C1 |->
%D    C2 C3 |->
%D    A0 A3 harrownodes nil 20 nil |->
%D    A2 B1 harrownodes nil 20 nil |-> sl^
%D    A2 B1 harrownodes nil 20 nil <-| sl_
%D    B0 B3 harrownodes nil 20 nil <-|
%D    B2 C1 harrownodes nil 20 nil <-| sl^
%D    B2 C1 harrownodes nil 20 nil |-> sl_
%D    C0 C3 harrownodes nil 20 nil |->
%D    A0 A2 -> .plabel= l    α
%D    A1 A3 -> .plabel= r Σ_fα
%D    A2 B0 -> .plabel= l \sm{f\\(Σ_f^♯)g}
%D    A3 B1 -> .plabel= r \sm{(Σ_f^♭)f\\g}
%D    B0 B2 -> .plabel= l f^*β
%D    B1 B3 -> .plabel= r    β
%D    B2 C0 -> .plabel= l \sm{(Π_f^♭)h\\k}
%D    B3 C1 -> .plabel= r \sm{h\\(Π_f^♯)k}
%D    C0 C2 -> .plabel= l    γ
%D    C1 C3 -> .plabel= r Π_fγ
%D    D0 D1 -> sl^^ .plabel= a Σ_f
%D    D0 D1 <-      .plabel= m f^*
%D    D0 D1 -> sl__ .plabel= b Π_f
%D    E2 E3  -> .plabel= a f
%D ))
%D enddiagram
%D
$$\pu
  \diag{adjs-generic}
$$

\newpage

%     _       _  _                           _       
%    / \   __| |(_)   __ _ _   _  __ _ _ __ | |_ ___ 
%   / _ \ / _` || |  / _` | | | |/ _` | '_ \| __/ __|
%  / ___ \ (_| || | | (_| | |_| | (_| | | | | |_\__ \
% /_/   \_\__,_|/ |  \__, |\__,_|\__,_|_| |_|\__|___/
%             |__/      |_|                          
%
% «adjoints-quants»  (to ".adjoints-quants")

Quantifiers as adjoints to adding a variable:

\bsk

%:                
%:                [\Pxy]^1        
%:                ::::::α         
%:                  \Qxy  
%:                --------
%:  \Exy\Pxy      \Exy\Qxy
%:  ----------------------1
%:  \Exy\Qxy
%:  
%:  ^Sigmapi-F
%:
%:
%:    \Qxy
%:  --------
%:  \Exy\Qxy
%:  :::
%:  \Rx
%:
%:  ^Sigmapi-transposeleft
%:
%:             [\Qxy]^1
%:              :::f
%:  \Exy\Qxy    \Rx
%:  ---------------1
%:  \Rx
%:
%:  ^Sigmapi-transposeright
%:
%:             
%:  \Rx
%:  :::β
%:  \Sx
%:
%:  ^pistar-F
%:
%:
%:    \Sx
%:  ::::::::k
%:  \Fay\Txy
%:  --------
%:  \Txy
%:
%:  ^Pipi-transposeleft
%:
%:       [\Sx]^1
%:       ::::
%:  \Sx  \Txy
%:  ---------
%:  \Fay\Txy
%:
%:  ^Pipi-transposeright
%:  
%:         
%:         
%:  \Fay\Txy    [\Txy]^1
%:  --------    ::::γ
%:  \Txy        \Uxy
%:  ----------------
%:  \Fay\Uxy
%:  
%:  ^Pipi-F
%:

\def\Pxy{Pxy}
\def\Qxy{Qxy}
\def\Rx {Rx}
\def\Sx {Sx}
\def\Txy{Txy}
\def\Uxy{Uxy}
\def\Exy{∃y.}
\def\Fay{∀y.}

%D diagram adjs-pi*
%D 2Dx     100    +40
%D 2D  100 A0 |-> A1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 A2 |-> A3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 B0 <-| B1
%D 2D      |       |
%D 2D      |  <-|  |
%D 2D      |       |
%D 2D  +25 B2 <-| B3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 C0 |-> C1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 C2 |-> C3
%D 2D
%D 2D  +20 D0 <=> D1
%D 2D
%D 2D  +20 E0 |-> E1
%D 2D  +10 E2 --> E3
%D 2D
%D ren A0 A1 ==> \Pxy \Exy\Pxy
%D ren A2 A3 ==> \Qxy \Exy\Qxy
%D ren B0 B1 ==> \Rx  \Rx
%D ren B2 B3 ==> \Sx  \Sx
%D ren C0 C1 ==> \Txy \Fay\Txy
%D ren C2 C3 ==> \Uxy \Fay\Uxy
%D ren D0 D1 ==> 𝐛P(X{×}Y)  𝐛P(X)
%D ren E0 E1 ==> (x,y) x
%D ren E2 E3 ==> X{×}Y X
%D
%D (( A0 A1 |->
%D    A2 A3 |->
%D    B0 B1 <-|
%D    B2 B3 <-|
%D    C0 C1 |->
%D    C2 C3 |->
%D    A0 A3 harrownodes nil 20 nil |->
%D    A2 B1 harrownodes nil 20 nil |-> sl^
%D    A2 B1 harrownodes nil 20 nil <-| sl_
%D    B0 B3 harrownodes nil 20 nil <-|
%D    B2 C1 harrownodes nil 20 nil <-| sl^
%D    B2 C1 harrownodes nil 20 nil |-> sl_
%D    C0 C3 harrownodes nil 20 nil |->
%D    A0 A2 -> .plabel= l    α
%D    A1 A3 -> .plabel= r Σ_πα
%D    A2 B0 -> .plabel= l \sm{f\\(Σ_π^♯)g}
%D    A3 B1 -> .plabel= r \sm{(Σ_π^♭)f\\g}
%D    B0 B2 -> .plabel= l π^*β
%D    B1 B3 -> .plabel= r    β
%D    B2 C0 -> .plabel= l \sm{(Π_π^♭)h\\k}
%D    B3 C1 -> .plabel= r \sm{h\\(Π_π^♯)k}
%D    C0 C2 -> .plabel= l    γ
%D    C1 C3 -> .plabel= r Π_πγ
%D    D0 D1 -> sl^^ .plabel= a Σ_π
%D    D0 D1 <-      .plabel= m π^*
%D    D0 D1 -> sl__ .plabel= b Π_π
%D    E0 E1 |->
%D    E2 E3  -> .plabel= a π
%D
%D    A1 A3 midpoint relplace  55 0     Σ_πα:=\pdedscale{0.55}{Sigmapi-F}
%D    A3 B1 midpoint relplace  55 0 (Σ_π^♭)f:=\pdedscale{0.55}{Sigmapi-transposeright}
%D    A2 B0 midpoint relplace -50 0 (Σ_π^♯)g:=\pdedscale{0.45}{Sigmapi-transposeleft}
%D    B0 B2 midpoint relplace -50 0     π^*β:=\pdedscale{0.55}{pistar-F}
%D    B2 C0 midpoint relplace -50 0 (Π_π^♭)h:=\pdedscale{0.40}{Pipi-transposeleft}
%D    B3 C1 midpoint relplace  50 0 (Π_π^♯)k:=\pdedscale{0.40}{Pipi-transposeright}
%D    C1 C3 midpoint relplace  55 0     Π_πγ:=\pdedscale{0.55}{Pipi-F}
%D ))
%D enddiagram
%D
\pu
\phantom{a}
\hspace{-20pt}
$
  \diag{adjs-pi*}
$

\newpage

%     _       _  _         
%    / \   __| |(_)  _____ 
%   / _ \ / _` || | |_____|
%  / ___ \ (_| || | |_____|
% /_/   \_\__,_|/ |        
%             |__/         
%
% «adjoints-equal»  (to ".adjoints-equal")

Equality as an adjoint to collapsing two variables:

\bsk

\def\exx {x{=}x}
\def\exxp{x{=}x'}
\def\Px  {Px}
\def\Qx  {Qx}
\def\Rxx {Rxx}
\def\Rxxp{Rxx'}
\def\Sxx {Sxx}
\def\Sxxp{Sxx'}
\def\Tx  {Tx}
\def\Ux  {Ux}



%:
%:               \exxp∧\Px
%:               ---------
%:  \exxp∧\Px    \Px
%:  ---------    :::α
%:  \exxp        \Qx
%:  ----------------
%:  \exxp∧\Qx
%:  
%:  ^SigmaD-F
%:
%:
%:  ----
%:  \exx   \Qx    [\exxp∧\Qx]^1
%:  ----------    :::::::::::g
%:  \exx∧\Qx       \Rxxp
%:  --------------------[x':=x];1
%:  \Rxx
%:
%:  ^SigmaD-transposeleft
%:
%:             \exxp∧\Qx
%:             ---------
%:  \exxp∧\Qx    \Qx
%:  ---------    ::::f
%:  \exxp        \Rxx
%:  -----------------
%:  \Rxxp
%:
%:  ^SigmaD-transposeright
%:
%:
%:
%:             
%:         [\Rxxp]^1
%:         :::::::::β
%:  \Rxx   \Sxxp
%:  ------------[x':=x];1
%:  \Sxx
%:
%:  ^Dstar-F
%:
%:  
%:  
%:  
%:                 [\Sxxp]^1
%:                 :::::::::h
%:          \Sxx   \exxp⊸\Tx
%:  ----    ----------------[x':=x];1
%:  \exx    \exx⊸\Tx
%:  ----------------
%:  \Tx
%:
%:  ^PiD-transposeleft
%:
%:                        [\Sxx]^1
%:                        ::::::::k
%:                          \Tx
%:                       --------1
%:           [\exxp]^2   \Sxx⊸\Tx
%:           --------------------
%:  \Sxxp    \Sxxp⊸\Tx
%:  ------------------
%:       \Tx
%:  ---------2
%:  \exxp⊸\Tx
%:
%:  ^PiD-transposeright
%:  
%:  [\exxp]^1   \exxp⊸\Tx
%:  ---------------------
%:     \Tx
%:     :::γ
%:     \Ux
%:  ---------1
%:  \exxp⊸\Ux
%:  
%:  ^PiD-F
%:


%D diagram adjs-Delta*
%D 2Dx     100    +40
%D 2D  100 A0 |-> A1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 A2 |-> A3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 B0 <-| B1
%D 2D      |       |
%D 2D      |  <-|  |
%D 2D      |       |
%D 2D  +25 B2 <-| B3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 C0 |-> C1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 C2 |-> C3
%D 2D
%D 2D  +20 D0 <=> D1
%D 2D
%D 2D  +20 E0 |-> E1
%D 2D  +10 E2 --> E3
%D 2D
%D ren A0 A1 ==> \Px \exxp∧\Px
%D ren A2 A3 ==> \Qx \exxp∧\Qx
%D ren B0 B1 ==> \Rxx \Rxxp
%D ren B2 B3 ==> \Sxx \Sxxp
%D ren C0 C1 ==> \Tx \exxp⊸\Tx
%D ren C2 C3 ==> \Ux \exxp⊸\Ux
%D ren D0 D1 ==> 𝐛P(X)  𝐛P(X{×}X)
%D ren E0 E1 ==> x (x,x')
%D ren E2 E3 ==> X X{×}X
%D
%D (( A0 A1 |->
%D    A2 A3 |->
%D    B0 B1 <-|
%D    B2 B3 <-|
%D    C0 C1 |->
%D    C2 C3 |->
%D    A0 A3 harrownodes nil 20 nil |->
%D    A2 B1 harrownodes nil 20 nil |-> sl^
%D    A2 B1 harrownodes nil 20 nil <-| sl_
%D    B0 B3 harrownodes nil 20 nil <-|
%D    B2 C1 harrownodes nil 20 nil <-| sl^
%D    B2 C1 harrownodes nil 20 nil |-> sl_
%D    C0 C3 harrownodes nil 20 nil |->
%D    A0 A2 -> .plabel= l    α
%D    A1 A3 -> .plabel= r Σ_Δα
%D    A2 B0 -> .plabel= l \sm{f\\(Σ_Δ^♯)g}
%D    A3 B1 -> .plabel= r \sm{(Σ_Δ^♭)f\\g}
%D    B0 B2 -> .plabel= l Δ^*β
%D    B1 B3 -> .plabel= r    β
%D    B2 C0 -> .plabel= l \sm{(Π_Δ^♭)h\\k}
%D    B3 C1 -> .plabel= r \sm{h\\(Π_Δ^♯)k}
%D    C0 C2 -> .plabel= l    γ
%D    C1 C3 -> .plabel= r Π_Δγ
%D    D0 D1 -> sl^^ .plabel= a Σ_Δ
%D    D0 D1 <-      .plabel= m Δ^*
%D    D0 D1 -> sl__ .plabel= b Π_Δ
%D    E0 E1 |->
%D    E2 E3  -> .plabel= a Δ
%D
%D    A1 A3 midpoint relplace  65 0     Σ_Δα:=\pdedscale{0.55}{SigmaD-F}
%D    A3 B1 midpoint relplace  65 0 (Σ_Δ^♭)f:=\pdedscale{0.55}{SigmaD-transposeright}
%D    A2 B0 midpoint relplace -65 0 (Σ_Δ^♯)g:=\pdedscale{0.45}{SigmaD-transposeleft}
%D    B0 B2 midpoint relplace -65 0     Δ^*β:=\pdedscale{0.55}{Dstar-F}
%D    B2 C0 midpoint relplace -65 0 (Π_Δ^♭)h:=\pdedscale{0.40}{PiD-transposeleft}
%D    B3 C1 midpoint relplace  65 0 (Π_Δ^♯)k:=\pdedscale{0.40}{PiD-transposeright}
%D    C1 C3 midpoint relplace  65 0     Π_Δγ:=\pdedscale{0.55}{PiD-F}
%D ))
%D enddiagram
%D
\pu
\phantom{a}
\hspace{-70pt}
$
  \diag{adjs-Delta*}
$

\newpage

%     _       _  _        __ 
%    / \   __| |(_)___   / _|
%   / _ \ / _` || / __| | |_ 
%  / ___ \ (_| || \__ \ |  _|
% /_/   \_\__,_|/ |___/ |_|  
%             |__/           
%
% «adjoints-f»  (to ".adjoints-f")

Adjoints to $(y:=f(x))^*$ can be built using quantifiers and equality:

\def\Px{Px}
\def\Qx{Qx}
\def\Ry{Ry}
\def\Sy{Sy}
\def\Rfx{Rfx}
\def\Sfx{Sfx}
\def\Tx{Txy}
\def\Ux{Uxy}
\def\Exx{∃x.}
\def\Fax{∃x.}
\def\fxy{fx{=}y}
\def\fxfx{fx{=}fx}

\bsk

%:                 
%:                 
%:                 
%:                            [\fxy∧\Px]^1
%:                            ------------
%:             [\fxy∧\Px]^1     \Px
%:             ------------     :::α         
%:                \fxy          \Qx    
%:                -----------------      
%:                  \fxy∧\Qx  
%:                  ------------
%:  \Exy\fxy∧\Px    \Exx\fxy∧\Qx
%:  ----------------------------1
%:  \Exx\fxy∧\Qx
%:  
%:  ^Sigmaf-F
%:
%:
%:  
%:  
%:  
%:                               [\fxy∧\Qx]^1
%:                               ------------
%:               [\fxy∧\Qx]^1    \Exx\fxy∧\Qx
%:  -----        ------------    :::
%:  \fxfx  \Qx   \fxy            \Ry
%:  ----------   -------------------
%:  \fxfx∧\Qx    \Rfx
%:  -----------------[y:=fx];1
%:  \Rfx
%:
%:  ^Sigmaf-transposeleft
%:
%:                         [\fxy∧\Qx]^1
%:                         ------------
%:           [\fxy∧\Qx]^1     \Qx
%:           ------------   ::::f
%:                \fxy      \Rfx
%:                --------------
%:  \Exx\fxy∧\Qx      \Ry
%:  ---------------------1
%:  \Ry
%:
%:  ^Sigmaf-transposeright
%:
%:             
%:             
%:         [\Ry]^1    
%:         :::::β    
%:  \Rfx   \Sy
%:  ----------[y:=fx]^1
%:  \Sfx
%:
%:  ^fstar-F
%:
%:
%:                      [\Sy]^1
%:                   ::::::::::::h
%:                   \Fay\fxy⊸\Tx
%:                   ------------
%:           \Sfx      \fxy⊸\Tx
%:  -----    ------------------[y:=fx];1
%:  \fxfx       \fxfx⊸\Tx
%:  ---------------------
%:  \Tx
%:
%:  ^Pif-transposeleft
%:
%:
%:      [\fxy]^1  [\Sy]^2
%:       ----------------
%:          Sfx
%:          :::
%:          Tx
%:       --------1
%:  \Sy  \fxy⊸\Tx
%:  -------------2
%:  \Fax\fxy⊸\Tx
%:
%:  ^Pif-transposeright
%:  
%:         
%:      [\fxy]^1  [\fxy⊸\Tx]^2
%:      ----------------------   
%:                  \Tx
%:                  :::γ 
%:                  \Ux
%:                --------1
%:  \Fax\fxy⊸\Tx   \fxy⊸\Ux
%:  -----------------------2
%:  \Fax\fxy⊸\Ux
%:  
%:  ^Pif-F
%:

\def\Pxy{Pxy}
\def\Qxy{Qxy}
\def\Rx {Rx}
\def\Sx {Sx}
\def\Tx {Tx}
\def\Ux {Ux}
\def\Exy{∃y.}
\def\Fay{∀y.}

%D diagram adjs-f*
%D 2Dx     100    +40
%D 2D  100 A0 |-> A1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 A2 |-> A3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 B0 <-| B1
%D 2D      |       |
%D 2D      |  <-|  |
%D 2D      |       |
%D 2D  +25 B2 <-| B3
%D 2D      |       |
%D 2D      |  <->  |
%D 2D      |       |
%D 2D  +25 C0 |-> C1
%D 2D      |       |
%D 2D      |  |->  |
%D 2D      |       |
%D 2D  +25 C2 |-> C3
%D 2D
%D 2D  +20 D0 <=> D1
%D 2D
%D 2D  +20 E0 |-> E1
%D 2D  +10 E2 --> E3
%D 2D
%D ren A0 A1 ==> \Px \Exx\fxy∧\Px
%D ren A2 A3 ==> \Qx \Exx\fxy∧\Qx
%D ren B0 B1 ==> \Rfx  \Ry
%D ren B2 B3 ==> \Sfx  \Sy
%D ren C0 C1 ==> \Tx \Exx\fxy⊸\Tx
%D ren C2 C3 ==> \Ux \Exx\fxy⊸\Ux
%D ren D0 D1 ==> 𝐛P(X)  𝐛P(Y)
%D ren E0 E1 ==> x  fx
%D ren E2 E3 ==> X  Y
%D
%D (( A0 A1 |->
%D    A2 A3 |->
%D    B0 B1 <-|
%D    B2 B3 <-|
%D    C0 C1 |->
%D    C2 C3 |->
%D    A0 A3 harrownodes nil 20 nil |->
%D    A2 B1 harrownodes nil 20 nil |-> sl^
%D    A2 B1 harrownodes nil 20 nil <-| sl_
%D    B0 B3 harrownodes nil 20 nil <-|
%D    B2 C1 harrownodes nil 20 nil <-| sl^
%D    B2 C1 harrownodes nil 20 nil |-> sl_
%D    C0 C3 harrownodes nil 20 nil |->
%D    A0 A2 -> .plabel= l    α
%D    A1 A3 -> .plabel= r Σ_πα
%D    A2 B0 -> .plabel= l \sm{f\\(Σ_π^♯)g}
%D    A3 B1 -> .plabel= r \sm{(Σ_π^♭)f\\g}
%D    B0 B2 -> .plabel= l π^*β
%D    B1 B3 -> .plabel= r    β
%D    B2 C0 -> .plabel= l \sm{(Π_π^♭)h\\k}
%D    B3 C1 -> .plabel= r \sm{h\\(Π_π^♯)k}
%D    C0 C2 -> .plabel= l    γ
%D    C1 C3 -> .plabel= r Π_πγ
%D    D0 D1 -> sl^^ .plabel= a Σ_f
%D    D0 D1 <-      .plabel= m f^*
%D    D0 D1 -> sl__ .plabel= b Π_f
%D    E0 E1 |->
%D    E2 E3  -> .plabel= a π
%D
%D    A1 A3 midpoint relplace  70 0     Σ_πα:=\pdedscale{0.40}{Sigmaf-F}
%D    A3 B1 midpoint relplace  70 0 (Σ_π^♭)f:=\pdedscale{0.40}{Sigmaf-transposeright}
%D    A2 B0 midpoint relplace - 0 (Σ_π^♯)g:=\pdedscale{0.40}{Sigmaf-transposeleft}
%D    B0 B2 midpoint relplace -50 0     π^*β:=\pdedscale{0.55}{fstar-F}
%D    B2 C0 midpoint relplace -60 0 (Π_π^♭)h:=\pdedscale{0.40}{Pif-transposeleft}
%D    B3 C1 midpoint relplace  55 0 (Π_π^♯)k:=\pdedscale{0.40}{Pif-transposeright}
%D    C1 C3 midpoint relplace  70 0     Π_πγ:=\pdedscale{0.40}{Pif-F}
%D ))
%D enddiagram
%D
\pu
\phantom{a}
\hspace{-75pt}
$
  \diag{adjs-f*}
$



%\printbibliography

\end{document}

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% <make>

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# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020seelyhyp-poster veryclean
make -f 2019.mk STEM=2020seelyhyp-poster pdf

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