Remark 4.6.1.9. Let $\{ \operatorname{\mathcal{C}}_{i} \} _{i \in I}$ be a collection of $\infty $-categories having a product $\operatorname{\mathcal{C}}= \prod _{i \in I} \operatorname{\mathcal{C}}_{i}$. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, which we identify with collections $\{ X_ i \in \operatorname{\mathcal{C}}_ i \} _{i \in I}$ and $\{ Y_ i \in \operatorname{\mathcal{C}}_ i \} _{i \in I}$, respectively. Then there is a canonical isomorphism of simplicial sets
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \prod _{i \in I} \operatorname{Hom}_{\operatorname{\mathcal{C}}_ i}( X_ i, Y_ i ). \]