Remark 4.8.2.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $n$ be an integer. Suppose that $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{D}}$ is locally $n$-truncated: that is, for every pair of objects $\overline{X}, \overline{Y} \in \operatorname{\mathcal{D}}$, the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(\overline{X}, \overline{Y})$ is $n$-truncated. To prove this, we can use the essential surjectivity of $F$ to reduce to the case where $\overline{X} =F(X)$ and $\overline{Y} = F(Y)$ for some objects $X,Y \in \operatorname{\mathcal{C}}$. In this case, the assertion follows from the observation that $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(\overline{X}, \overline{Y})$ is an $n$-truncation of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.
Conversely, if $\operatorname{\mathcal{D}}$ is locally $n$-truncated, then $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ if and only if it is essentially surjective and satisfies the following weaker version of condition $(2)$:
- $(2')$
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the map of Kan complexes
\[ F_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) ) \]is $(n+1)$-connective.