The category of relations from $A$ to $B$ is the category $\mathbf{Rel}\webleft (A,B\webright )$ defined by1
\[ \mathbf{Rel}\webleft (A,B\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathbf{Rel}\webleft (A,B\webright )_{\mathsf{pos}}, \]
where $\mathbf{Rel}\webleft (A,B\webright )_{\mathsf{pos}}$ is the posetal category associated to the poset $\mathbf{Rel}\webleft (A,B\webright )$ of Item 2 of Notation 6.1.1.1.4 and Chapter 9: Preorders, Definition 9.2.7.1.1.
1Here we choose to abuse notation by writing $\mathbf{Rel}\webleft (A,B\webright )$ instead of $\mathbf{Rel}\webleft (A,B\webright )_{\mathsf{pos}}$ 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 $B$.
