Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.6.4.19. Let $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be fully faithful functor of $\infty $-categories and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the induced functors

\[ G': \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{D}}_{/(G \circ F)} \quad \quad G'': \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{D}}_{(G \circ F)/} \]

are also fully faithful.

Proof. Let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{D}}$ be the essential image of $G$ (Definition 4.6.2.9), so that $G$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ (Corollary 4.6.2.19). By virtue of Corollary 4.6.4.18, the functors $G'$ and $G''$ restrict to equivalences

\[ \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{\mathcal{C}}'_{/(G \circ F)} \quad \quad \operatorname{\mathcal{C}}_{F/} \rightarrow \operatorname{\mathcal{C}}'_{(G \circ F)/} \]

We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $G: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is the inclusion of a full subcategory. In this case, the functors $G'$ and $G''$ are also the inclusions of full subcategories, hence fully faithful (Example 4.6.2.2). $\square$