Corollary 7.6.1.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then $\operatorname{\mathcal{C}}$ admits finite products if and only if it satisfies the following pair of conditions:
The $\infty $-category $\operatorname{\mathcal{C}}$ has a final object ${\bf 1}$.
The $\infty $-category $\operatorname{\mathcal{C}}$ admits pairwise products. That is, every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ have a product $X \times Y$ in $\operatorname{\mathcal{C}}$.
Proof. The necessity of conditions $(1)$ and $(2)$ is clear (see Example 7.6.1.8). Conversely, suppose that $(1)$ and $(2)$ are satisfied, and let $I$ be a finite set. We wish to show that $\operatorname{\mathcal{C}}$ admits $I$-indexed limits. We proceed by induction on the cardinality of $I$. If $I$ is empty, then the desired result follows from assumption $(1)$. If $I$ is a singleton, then the desired result is obvious (see Example 7.6.1.9). Otherwise, we can write $I$ as a disjoint union of proper subsets $I_{-}, I_{+} \subset I$. Our inductive hypothesis then guarantees that $\operatorname{\mathcal{C}}$ admits $I_{-}$-indexed limits and $I_{+}$-indexed limits. Combining assumption $(2)$ with Corollary 7.6.1.22, we deduce that $\operatorname{\mathcal{C}}$ admits limits indexed by $I = I_{-} \coprod I_{+}$. $\square$