Kerodon

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

Corollary 4.6.2.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. Then the induced map of cores $\operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ is also fully faithful.

Proof. Fix objects $X,Y \in \operatorname{\mathcal{C}}^{\simeq }$. Our assumption that $F$ is fully faithful guarantees that the induced map $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ is a homotopy equivalence of Kan complexes. By virtue of Proposition 4.6.2.7, $\theta $ restricts to a homotopy equivalence from the summand of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ spanned by the isomorphisms from $X$ to $Y$ to the summand of $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$ spanned by the isomorphisms from $F(X)$ to $F(Y)$. Unwinding the definitions, we conclude that $F^{\simeq }$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}^{\simeq }}( X, Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}^{\simeq }}( F(X), F(Y) )$. $\square$