Example 7.6.1.16 (Products in a Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category and let $\{ q_ i: Y \rightarrow Y_ i \} $ be a collection of $1$-morphisms in $\operatorname{\mathcal{C}}$. Then 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 $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$.
- $(2)$
For every object $X \in \operatorname{\mathcal{C}}$, horizontal composition with the $1$-morphisms $q_ i$ induces an equivalence of categories
\[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow {\prod }_{i \in I} \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Y_ i ). \]
This follows from the explicit description of pinched morphism spaces in $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ supplied by Example 4.6.5.13.