Definition 2.2.8.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\operatorname{\mathcal{H}}$ be an ordinary category, viewed as a $2$-category having only identity $2$-morphisms. We say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{H}}$ exhibits $\operatorname{\mathcal{H}}$ as a coarse homotopy category of $\operatorname{\mathcal{C}}$ if, for every ordinary category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors of ordinary categories from $\operatorname{\mathcal{H}}$ to $\operatorname{\mathcal{E}}$} \} \ar [d] \\ \{ \textnormal{Functors of $2$-categories from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{E}}$} \} . } \]