Example 7.6.1.13 (Homotopy Products). Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, and let $\{ q_ i: Y \rightarrow Y_ i \} _{i \in I}$ be a collection of morphisms in $\operatorname{\mathcal{C}}$. By virtue of Theorem 4.6.8.5 (and Proposition 4.6.9.19), the following conditions are equivalent:
- $(1)$
The morphisms $q_ i$ exhibit $Y$ as a product of the collection $\{ Y_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$.
- $(2)$
For every object $X \in \operatorname{\mathcal{C}}$, composition with the morphisms $q_ i$ determines a homotopy equivalence of Kan complexes
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow {\prod }_{i \in I} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y_ i )_{\bullet }. \]