The 2-category $\boldsymbol {\mathsf{Rel}}$ 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
\begin{align*} \mathbf{Rel}(S\mathbin {\diamond }R,T) & \cong \mathbf{Rel}(S,\operatorname {\mathrm{Ran}}_{R}(T)),\\ \mathbf{Rel}(R\mathbin {\diamond }U,V) & \cong \mathbf{Rel}(U,\operatorname {\mathrm{Rift}}_{R}(V)), \end{align*}
natural in $S\in \mathrm{Rel}(B,X)$, $T\in \mathrm{Rel}(A,X)$, $U\in \mathrm{Rel}(X,A)$, and $V\in \mathrm{Rel}(X,B)$.
