%% %% 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}