Proposition 4.8.2.18. Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{D}}$ is locally $n$-truncated. The following conditions are equivalent:
- $(1)$
The functor $F$ exhibits $\operatorname{\mathcal{D}}$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$, in the sense of Definition 4.8.2.9.
- $(2)$
For every locally $n$-truncated $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.
- $(3)$
For every locally $n$-truncated $\infty $-category $\operatorname{\mathcal{E}}$, precomposition with $F$ induces a bijection
\[ \pi _0( \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ). \]
Proof.
Without loss of generality we may assume that $n \geq -2$. We first show that $(1)$ implies $(2)$. By virtue of Corollary 4.8.2.17, we can assume without loss of generality that $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{E}}$ are $(n+2)$-coskeletal. In this case, $F$ factors (uniquely) as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \xrightarrow {F''} \operatorname{\mathcal{D}}$, where $F'$ exhibits $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ as a local $n$-truncation of $\operatorname{\mathcal{C}}$ (Proposition 4.8.2.13). Applying Remark 4.8.2.12, we see that $F''$ is an equivalence of $\infty $-categories. We may therefore replace $\operatorname{\mathcal{D}}$ by $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ and thereby reduce to the case where $F$ exhibits $\operatorname{\mathcal{D}}$ as an $(n+2)$-coskeleton of $\operatorname{\mathcal{C}}$. In this case, Proposition 3.5.3.17 guarantees that the precomposition functor
\[ \operatorname{Fun}( \operatorname{\mathcal{D}},\operatorname{\mathcal{E}}) \xrightarrow {\circ F} \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]
is an isomorphism of simplicial sets (and therefore an equivalence of $\infty $-categories).
The implication $(2) \Rightarrow (3)$ follows immediately from the definitions. We will complete the proof by showing that $(3)$ implies $(1)$. As before, we may assume that $\operatorname{\mathcal{D}}= \operatorname{cosk}_{n+2}(\operatorname{\mathcal{D}})$ is $(n+2)$-coskeletal from Proposition 4.5.1.22, so that $F$ factors (uniquely) as a composition $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}) \xrightarrow {F''} \operatorname{\mathcal{D}}$; we wish to show that the homotopy class $[F'']$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$. Since $\operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}})$ and $\operatorname{\mathcal{D}}$ are locally $n$-truncated, it will suffice to show that for every locally $n$-truncated $\infty $-category $\operatorname{\mathcal{E}}$, the horizontal map in the diagram
\[ \xymatrix@R =50pt@C=50pt{ \pi _0(\operatorname{Fun}( \operatorname{\mathcal{D}}, \operatorname{\mathcal{E}})^{\simeq } ) \ar [rr] \ar [dr] & & \pi _0(\operatorname{Fun}( \operatorname{cosk}_{n+2}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}})^{\simeq }) \ar [dl] \\ & \pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})^{\simeq } ) & } \]
is a bijection. This is clear, since the vertical maps are bijections.
$\square$