The category of relations from $A$ to $B$ is the category $\mathbf{Rel}(A,B)$ defined by1
\[ \mathbf{Rel}(A,B)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathbf{Rel}(A,B)_{\mathsf{pos}}, \]
where $\mathbf{Rel}(A,B)_{\mathsf{pos}}$ is the posetal category associated to the poset $\mathbf{Rel}(A,B)$ of Item 2 of Notation 8.1.1.1.4 and Chapter 11: Categories, Definition 11.2.7.1.1.
- 1Here we choose to abuse notation by writing $\mathbf{Rel}(A,B)$ instead of $\mathbf{Rel}(A,B)_{\mathsf{pos}}$ for the posetal category of relations from $A$ to $B$, even though the same notation is used for the poset of relations from $A$ to $B$.
