Construction 2.2.8.12 (The Homotopy Category of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a category $\operatorname{hPith}(\operatorname{\mathcal{C}})$ as follows:
The objects of $\operatorname{hPith}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.
If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, then $\operatorname{Hom}_{\operatorname{hPith}(\operatorname{\mathcal{C}})}(X,Y)$ is the set of isomorphism classes of objects in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. If $f: X \rightarrow Y$ is a $1$-morphism from $X$ to $Y$, we typically denote its isomorphism class by $[f] \in \operatorname{Hom}_{ \operatorname{hPith}(\operatorname{\mathcal{C}}) }(X,Y)$.
The composition law on $\operatorname{hPith}(\operatorname{\mathcal{C}})$ is determined by the requirement that $[g] \circ [f] = [g \circ f]$ for every pair of composable $1$-morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ (this composition law is associative by virtue of the existence of the associativity contraints of the $2$-category $\operatorname{\mathcal{C}}$).
For every object $Y \in \operatorname{\mathcal{C}}$, the identity morphism from $Y$ to itself in $\operatorname{hPith}(\operatorname{\mathcal{C}})$ is the isomorphism class of the identity morphism $\operatorname{id}_{Y}$ in $\operatorname{\mathcal{C}}$. For $1$-morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$, the identities
\[ [\operatorname{id}_ Y] \circ [f] = [f] \quad \quad [g] \circ [\operatorname{id}_ Y] = [g] \]follow from the existence of left and right unit constraints (see Construction 2.2.1.11).
We will refer to $\operatorname{hPith}(\operatorname{\mathcal{C}})$ as the homotopy category of $\operatorname{\mathcal{C}}$.