Definition 4.8.2.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. We say that $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ if the following conditions are satisfied:
- $(1)$
The functor $F$ is essentially surjective (Definition 4.6.2.12).
- $(2)$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of Kan complexes
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]exhibits $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ as an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, in the sense of Definition 3.5.7.19.