Construction 2.2.8.2 (The Coarse Homotopy Category of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:
The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.
If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ is the set of connected components of the simplicial set $\operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )$.
For objects $X$, $Y$, and $Z$ of $\operatorname{\mathcal{C}}$, the composition of morphisms in $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is given by the map
\begin{eqnarray*} \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,Z) \times \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) & = & \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) ) \times \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \\ & \simeq & \pi _0( \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) ) \\ & \xrightarrow {\circ } & \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) ) \\ & = & \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z). \end{eqnarray*}
We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the coarse homotopy category of $\operatorname{\mathcal{C}}$.