Example 1.4.0.7 (Coproducts of $\infty $-Categories). Let $\{ S_{\alpha } \} _{\alpha \in A}$ be a collection of simplicial sets parametrized by a set $A$, and let $S = \coprod _{\alpha \in A} S_{\alpha }$ denote their coproduct. For each $0 < i < n$, the restriction map
can be identified with the coproduct (formed in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$) of restriction maps $\theta _{\alpha }: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S_{\alpha }) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^ n_ i, S_{\alpha } )$; this follows from the connectedness of the simplicial sets $\Delta ^{n}$ and $\Lambda ^{n}_{i}$ (see Example 1.2.1.7 and Remark 1.2.4.5). It follows that $\theta $ is a surjection if and only if each $\theta _{\alpha }$ is a surjection. Allowing $n$ and $i$ to vary, we conclude that $S$ is an $\infty $-category if and only if each summand $S_{\alpha }$ is an $\infty $-category.