\[\newcommand\bm[1]{\boldsymbol{#1}}
\newcommand\EA{\pmb{\bm{\mathtt{EA}}}}
\newcommand\BA{\pmb{\bm{\mathtt{BA}}}}
\newcommand\DL{\pmb{\bm{\mathtt{DL}}}}
\newcommand\Rng{\pmb{\bm{\mathbf{Rng}}}}
\newcommand\Cpu{\pmb{\bm{\mathbf{C}}^\ast}_{\hspace{-1ex}\rm PU}}
\newcommand\Hilb{\pmb{\bm{\mathbf{Hilb}}}}
\newcommand\HilbShort{\pmb{\bm{\mathbf{Hilb}_{\text{\tiny short}}}}}
\newcommand\BanShort{\pmb{\bm{\mathbf{Ban}_{\text{\tiny short}}}}}
\newcommand\Ban{\pmb{\bm{\mathbf{Ban}}}}
\newcommand\Bounded{\mathop{ {\mathcal{B}}}}
\newcommand\DM{\mathop{ {\mathcal{DM}}}}
\newcommand\Tr{\mathop{ {\mathrm{Tr}}}}
\newcommand\FProd[2][{}]{\mathord{\mathrel{\rm FProd}\parent{#2}}}
\newcommand\PCM{\pmb{\bm{\mathtt{PCM}}}}
\newcommand\PCMod{\pmb{\bm{\mathtt{PCMod}}}}
\newcommand\EMonNorm{\pmb{\bm{\mathtt{EMonNorm}}}}
\newcommand\EMod[1]{\pmb{\bm{\mathtt{EMod}}}_{#1}}
\newcommand\EMon{\pmb{\bm{\mathtt{EMon}}}}
\newcommand\TricoCanc{\pmb{\bm{\mathtt{TricoCanc}}}}
\newcommand\D[1]{\mathop{\mathcal{D}_{#1}}}
\newcommand\Endop[1]{\mathop{\mathrm{End}_{#1}}}
\newcommand\G{\mathop{\mathcal{G}}}
\newcommand\M[1][\mathbb{B}]{M_{#1}}
\newcommand\Pred{\mathop{\mathrm {Pred}}}
\newcommand\Stat{\mathop{\mathrm {Stat}}}
\newcommand\Conv[1]{\mathrm{Conv}_{#1}}
\newcommand\Comma[2]{#1\downarrow#2}
\newcommand\lub{\bigvee}
\newcommand\glb{\bigwedge}
\newcommand\restrict[1]{\left.\vphantom{\int}\right\lvert_{#1}}
\newcommand{\map}[1]{\require{HTML} \style{display: inline-block; border-style: none}{\mmlToken{mglyph}[src="https://younesse.net/assets/Slides/M2-Internship/Pictures/Maps/map_#1.svg" width="12mm" height="12mm" valign="-2mm" alt="Shuffle map" class="noborder" style="margin-top:15px"]{}}}
\newcommand{\ovee}{\require{HTML} \style{display: inline-block; border-style: none}{ \mmlToken{mglyph}[src="https://younesse.net/assets/Slides/M2-Internship/Pictures/ovee.svg" alt="ovee" class="noborder ovee"]{}}}
\newcommand\dom{\mathop{\mathrm{dom}}}
\newcommand\supp{\mathop{\mathrm{supp}}}
\newcommand{\closure}[2][3]{ {}\mkern#1mu\overline{\mkern-#1mu#2}}
\newcommand\restr[1]{\raisebox{-.5ex}{$|$}_{#1}}
\newcommand\yoneda{ {\bf y}}
\newcommand\oppositeName{ {\rm op}}
\newcommand\opposite[1]{ {#1}^\oppositeName}
\newcommand\id[1][{}]{ {\rm id}_{#1}}
\newcommand\Id[1][{}]{ {\rm Id}_{#1}}
\newcommand\Cat[1]{\mathcal{#1}\/}
\newcommand\Category[1]{ {\mathbf{ #1}}}
\newcommand\Set{\Category{Set}}
\newcommand\tensor{\otimes}
\newcommand\unit{\mathbb I}
\newcommand\carrier[1]{\underline{#1}}
\newcommand\Comma[2]{#1\downarrow#2}
\newcommand\lub{\bigvee}
\newcommand\glb{\bigwedge}
\newcommand\restrict[1]{\left.\vphantom{\int}\right\lvert_{#1}}
\newcommand\catA{\Category A}
\newcommand\catB{\Category B}
\newcommand\catC{\Category C}
\newcommand\catD{\Category D}
\newcommand\catE{\Category E}
\newcommand\Psh[1]{\widehat{#1}}
\newcommand\PshStar[1]{\widehat{#1}}
\newcommand{\Pfin}{\mathop{ {\mathcal P}_{\rm fin}}\nolimits}
\newcommand\eqdef{≝}
\newcommand\ie{\emph{i.e.~}}
\newcommand\conflict{\mathrel{\sim\joinrel\sim}}
\newcommand\Cocomp{\mathbf{Cocomp}}
\newcommand\const[1]{\Delta_{#1}}
\newcommand\Lan[2]{\mathop{\mathrm{Lan}}_{#1}(#2)}
\newcommand\Ran[2]{\mathop{\mathrm{Ran}}_{#1}(#2)}
\newcommand\Nerve[1]{\mathrm{N}_{#1}}
\newcommand{\dinat}{\stackrel{\bullet}{\longrightarrow}}
\newcommand{\End}[2][c]{\int_{#1} #2(c, c)}
\newcommand{\Coend}[2][c]{\int^{#1} #2(c, c)}
\newcommand{\obj}[1]{\vert #1 \vert}
\newcommand{\elem}[1]{\int #1}
\newcommand{\tens}[2]{#1 \cdot #2}
\newcommand{\cotens}[2]{#2^{#1}}
\newcommand{\Nat}{\mathop{\rm Nat}\nolimits}
\newcommand{\colim}{\mathop{\rm colim}\nolimits}
\newcommand{\cancolimAC}[1]{\big((i/#1)\stackrel{U}{→} \catA \stackrel{i}{→} \catC\big)}
\newcommand{\canlimAC}[1]{\big((#1\backslash i)\stackrel{U}{→} \catA \stackrel{i}{→} \catC\big)}
\newcommand\pair[2]{\left<{#1}, {#2}\right>}
\newcommand\triple[3]{\anglebrackets{ {#1}, {#2}, {#3}}}
\newcommand\anglebrackets[1]{\left<{#1}\right>}
\newcommand\set[1]{\left\{#1\right\}}
\newcommand\suchthat{\middle\vert}
\newcommand\xto\xrightarrow
\newcommand\xfrom\xleftarrow
\newcommand\parent[1]{\left({#1}\right)}
\newcommand\Hom[2][{}]{\mathord{\mathrel{\rm Hom}_{#1}\parent{#2}}}
\newcommand{\bigovee}{
\mathop{
\mathchoice{\dobigovee\Large}
{\dobigovee\large}
{\dobigovee\normalsize}
{\dobigovee\small}
}
}
\newcommand{\dobigovee}[1]{\vcenter{#1\kern.2ex\hbox{$\ovee$}\kern.2ex}}
\def\Bra#1{\left\langle#1\right|}
\def\Ket#1{\left|#1\right\rangle}
\newcommand{\EM}{\mathop{\cal EM}\nolimits}
\newcommand{\Kl}{\mathop{ {\cal K}\hspace{-.1em}\ell}\nolimits}
\newcommand{\KlN}{\mathop{ {\cal K}\hspace{-.1em}\ell_\mathbb{N}}\nolimits}
\xymatrix @!0 @R=2pc @C=4pc {
\def\adj{\ar@{}[r]|<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<{\top}}
\opposite {(\EMod{\M})} \; \ar@/^1em/[rr]^{\Hom[\EMod{\M}]{-, \, \M}} \adj && \ar@/^1em/[ll]^{\Hom[\Conv {\M}]{-, \, \M}} \; \Conv {\M} ≅ \EM(\D {\M})\\
& 𝔹 \ar@/^1em/[lu]^-{\Pred ≝ \Hom[𝔹]{-, \, 1+1}\qquad} \ar@/_1em/[ru]_-{\qquad \Stat ≝ \Hom[𝔹]{1, \, -}} &
}\]