Let $A$ and $B$ be sets and let $R,S\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be relations.
Proof of Proposition 6.1.1.1.5.
Item 1: End Formula for the Set of Inclusions of Relations
Unwinding the expression inside the end on the right hand side, we have
\[ \int _{a\in A}\int _{b\in B}\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{a},S^{b}_{a}\webright )\cong \begin{cases} \text{pt}& \text{if, for each $a\in A$ and each $b\in B$,}\\ & \text{we have $\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{a},S^{b}_{a}\webright )\cong \text{pt}$}\\ \text{Ø}& \text{otherwise.}\end{cases} \]
Since we have $\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{a},S^{b}_{a}\webright )=\webleft\{ \mathsf{true}\webright\} \cong \text{pt}$ exactly when $R^{b}_{a}=\mathsf{false}$ or $R^{b}_{a}=S^{b}_{a}=\mathsf{true}$, we get
\[ \int _{a\in A}\int _{b\in B}\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{a},S^{b}_{a}\webright )\cong \begin{cases} \text{pt}& \text{if, for each $a\in A$ and each $b\in B$,}\\ & \text{if $a\sim _{R}b$, then $a\sim _{S}b$,}\\ \text{Ø}& \text{otherwise.}\end{cases} \]
On the left hand-side, we have
\[ \textup{Hom}_{\mathbf{Rel}\webleft (A,B\webright )}\webleft (R,S\webright )\cong \begin{cases} \text{pt}& \text{if $R\subset S$,}\\ \text{Ø}& \text{otherwise.}\end{cases} \]
It is then clear that the conditions for each set to evaluate to $\text{pt}$ (up to isomorphism) are equivalent, implying that those two sets are isomorphic.
