Kerodon

Warning 7.6.1.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. if $\{ q_ i: Y \rightarrow Y_ i \} _{i \in I}$ is a collection of morphisms of $\operatorname{\mathcal{C}}$ which exhibits $Y$ as a product of the collection of objects $\{ Y_ i \} _{i \in I}$ in the $\infty $-category $\operatorname{\mathcal{C}}$, then the collection of homotopy classes $\{ [q_ i]: Y \rightarrow Y_ i \} _{i \in I}$ exhibits $Y$ as a product of the collection $\{ Y_ i \} _{i \in I}$ in the ordinary category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. The converse holds if the collection $\{ Y_ i \} _{i \in I}$ admits a products in the $\infty $-category $\operatorname{\mathcal{C}}$. However, the converse need not hold in general, even in the special case where the set $I$ is empty: see Warning 4.6.7.18.