Corollary 4.6.2.23. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a fully faithful functor of $\infty $-categories. Then, for every simplicial set $K$, the induced map $\operatorname{Fun}(K, \operatorname{\mathcal{C}}) \xrightarrow {F \circ } \operatorname{Fun}(K, \operatorname{\mathcal{D}})$ is also fully faithful.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Corollary 4.6.2.22, we can replace $\operatorname{\mathcal{C}}$ by its essential image and thereby reduce to the case where $F: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{D}}$ is the inclusion of a full subcategory. In this case, the induced map $\operatorname{Fun}(K, \operatorname{\mathcal{C}}) \xrightarrow {F \circ } \operatorname{Fun}(K,\operatorname{\mathcal{D}})$ is also the inclusion of a full subcategory, and therefore automatically fully faithful (Example 4.6.2.2). $\square$