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%%
%% An UIT Edition example
%%
%% Example 10-01-4 on page 199.
%%
%% Copyright (C) 2010 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
%%
%% ====
% Show page(s) 1
%%
\documentclass[]{exaarticle}
\pagestyle{empty}
\setlength\textwidth{375.57637pt}
\usepackage[utf8]{inputenc}
\AtBeginDocument{\setlength\parindent{0pt}}
\StartShownPreambleCommands
\usepackage{amsmath,amssymb,amsopn,pst-node}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Mod}{Mod}
\DeclareMathOperator{\obj}{obj}
\StopShownPreambleCommands
\begin{document}
Let $\mathcal{C}:=\mathcal{D}:=\Mod_A$, $M\in
\obj(\Mod_A)$ be fixed,
\begin{align*}
F:=\square\otimes_A M : \Mod_A &\rightarrow \Mod_A, \\
G:=\Hom_A(M,\square) : \Mod_A &\rightarrow \Mod_A.
\end{align*}
These two functors are adjoint, by
$h(X,Y)(\phi)(x)(m)\rightarrow\phi(x\otimes m)$:\\[10pt]
$\begin{psmatrix}[colsep=-0.3cm, rowsep=0pt]
\quad\phi{\;} & & {\;} h(X,Y)(\phi)\quad\\
\Hom_A(M \otimes_A X, Y) & & \Hom_A(X, \Hom_A(M, Y)) \\[1.5cm]
& L_A^2(M, X; Y)
\end{psmatrix}$
\ncline{|->}{1,1}{1,3}
\ncline[doubleline=true]{2,1}{2,3}\nbput{h(X,Y)}
\naput[nrot=:U,labelsep=0pt]{$\sim$}
\ncline[doubleline=true]{2,1}{3,2}\naput[nrot=:U,labelsep=0pt]{$\sim$}
\ncline[doubleline=true]{3,2}{2,3}\naput[nrot=:U,labelsep=0pt]{$\sim$}
\end{document}
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