The horizontal unit functor of $\mathsf{Rel}^{\mathsf{dbl}}$ is the functor
\[ \mathbb {1}^{\mathsf{Rel}^{\mathsf{dbl}}} \colon \mathsf{Rel}^{\mathsf{dbl}}_{0} \to \mathsf{Rel}^{\mathsf{dbl}}_{1} \]
of $\mathsf{Rel}^{\mathsf{dbl}}$ is the functor where
- Action on Objects. For each $A\in \text{Obj}\webleft (\mathsf{Rel}^{\mathsf{dbl}}_{0}\webright )$, we have
\[ \mathbb {1}_{A} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{A}\webleft (-_{1},-_{2}\webright ). \]
- Action on Morphisms. For each vertical morphism $f\colon A\to B$ of $\mathsf{Rel}^{\mathsf{dbl}}$, i.e. each map of sets $f$ from $A$ to $B$, the identity $2$-morphism
of $f$ is the inclusion
of Chapter 2: Constructions With Sets, Item 1 of Proposition 2.4.1.1.3.
