# Kerodon

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

Example 4.6.2.2. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory (Definition 4.1.2.15). Then the inclusion map $\iota : \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is fully faithful. In fact, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}'$, the inclusion $\iota$ induces an isomorphism of simplicial sets $\operatorname{Hom}_{\operatorname{\mathcal{C}}'}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.