Example 7.6.1.19 (Products in a Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category and let $\{ q_ i: Y \rightarrow Y_ i \} $ be a collection of morphisms in the underlying category of $\operatorname{\mathcal{C}}$ (that is, each $q_ i$ is a $0$-cycle of the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y, Y_ i)_{\ast }$). Using Example 4.6.5.15 (together with Exercise 3.2.2.22), we see that 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 $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.
- $(2)$
For every object $X \in \operatorname{\mathcal{C}}$, the map of chain complexes
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } \rightarrow {\prod }_{i \in I} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y_ i)_{\ast } \]induces an isomorphism on homology in degrees $\geq 0$.