6.3.13 Closedness of $\textbf{Rel}$

Proposition 6.3.13.1.1 ▶ Closedness of $\textbf{Rel}$

The $2$-category $\textbf{Rel}$ is a closed bicategory, there being, for each $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ and set $X$, a pair of adjunctions

witnessed by bijections

\begin{align*} \mathbf{Rel}\webleft (S\mathbin {\diamond }R,T\webright ) & \cong \mathbf{Rel}\webleft (S,\text{Ran}_{R}\webleft (T\webright )\webright ),\\ \mathbf{Rel}\webleft (R\mathbin {\diamond }U,V\webright ) & \cong \mathbf{Rel}\webleft (U,\text{Rift}_{R}\webleft (V\webright )\webright ), \end{align*}

natural in $S\in \mathrm{Rel}\webleft (B,X\webright )$, $T\in \mathrm{Rel}\webleft (A,X\webright )$, $U\in \mathrm{Rel}\webleft (X,A\webright )$, and $V\in \mathrm{Rel}\webleft (X,B\webright )$.

Proof of Proposition 6.3.13.1.1.
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