Warning 2.2.8.14. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{hPith}(\operatorname{\mathcal{C}})$ be the homotopy category of $\operatorname{\mathcal{C}}$, which we regard as a $2$-category having only identity $2$-morphisms. In general, there is no functor which directly relates $\operatorname{\mathcal{C}}$ to the homotopy category $\operatorname{hPith}(\operatorname{\mathcal{C}})$. Instead, there is a commutative diagram of $2$-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Pith}(\operatorname{\mathcal{C}}) \ar@ {^{(}->}[r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{hPith}(\operatorname{\mathcal{C}}) \ar [r] & \mathrm{h} \mathit{\operatorname{\mathcal{C}}}. } \]
Here the functor $\operatorname{hPith}(\operatorname{\mathcal{C}}) \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is bijective on objects and full: that is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{Hom}_{ \operatorname{hPith}(\operatorname{\mathcal{C}})}(X,Y) = \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq } ) \rightarrow \pi _0( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) = \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X,Y) \]
is surjective.