The $2$-category of relations is the locally posetal $2$-category $\textbf{Rel}$ where
- Objects. The objects of $\textbf{Rel}$ are sets.
- $\mathbf{Hom}$-Objects. For each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
\begin{align*} \textup{Hom}_{\textbf{Rel}}\webleft (A,B\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathbf{Rel}\webleft (A,B\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (\mathrm{Rel}\webleft (A,B\webright ),\subset \webright ).\end{align*}
- Identities. For each $A\in \text{Obj}\webleft (\textbf{Rel}\webright )$, the unit map
\[ \mathbb {1}^{\textbf{Rel}}_{A} \colon \text{pt}\to \mathbf{Rel}\webleft (A,A\webright ) \]
of $\textbf{Rel}$ at $A$ is defined by
\[ \text{id}^{\textbf{Rel}}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{A}\webleft (-_{1},-_{2}\webright ), \]where $\chi _{A}\webleft (-_{1},-_{2}\webright )$ is the characteristic relation of $A$ of Chapter 2: Constructions With Sets, Item 3 of Definition 2.4.1.1.1.
- Composition. For each $A,B,C\in \text{Obj}\webleft (\textbf{Rel}\webright )$, the composition map[1]
\[ \circ ^{\textbf{Rel}}_{A,B,C}\colon \mathbf{Rel}\webleft (B,C\webright )\times \mathbf{Rel}\webleft (A,B\webright )\to \mathbf{Rel}\webleft (A,C\webright ) \]
of $\textbf{Rel}$ at $\webleft (A,B,C\webright )$ is defined by
\[ S\mathbin {{\circ }^{\textbf{Rel}}_{A,B,C}}R \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}S\mathbin {\diamond }R \]for each $\webleft (S,R\webright )\in \textbf{Rel}\webleft (B,C\webright )\times \textbf{Rel}\webleft (A,B\webright )$, where $S\mathbin {\diamond }R$ is the composition of $S$ and $R$ of Chapter 6: Constructions With Relations, Definition 6.3.12.1.1.
