Proposition 7.6.1.12 (04Q7)

Proposition 7.6.1.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $X$. The following conditions are equivalent:

$(1)$: For every object $Y \in \operatorname{\mathcal{C}}$, there exists a product of $X \times Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
$(2)$: The forgetful functor $U: \operatorname{\mathcal{C}}_{ / Y} \rightarrow \operatorname{\mathcal{C}}$ admits a right adjoint.

If these conditions are satisfied, then the right adjoint of $U$ is given on objects by the construction $Y \mapsto X \times Y$.

Proof. If $Y$ is an object of $\operatorname{\mathcal{C}}$, then a product $X \times Y$ (if it exists) can be identified with a final object of the $\infty $-category $\operatorname{\mathcal{C}}_{/X} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$. The equivalence of $(1)$ and $(2)$ is therefore a special case of the criterion of Corollary 6.2.4.2. $\square$