Definition 4.8.4.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ be a functor of $\infty $-categories and let $n$ be an integer. We say that $F$ exhibits $\operatorname{\mathcal{C}}'$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}'$ is an $(n,1)$-category (Definition 4.8.1.8).
- $(2)$
For every $(n,1)$-category $\operatorname{\mathcal{D}}$, precomposition with $F$ induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors of $(n,1)$-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Functors of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} . } \]