The Stacks project

Definition 4.14.6. Suppose that $I$ is a set, and suppose given for every $i \in I$ an object $M_ i$ of the category $\mathcal{C}$. A product $\prod _{i\in I} M_ i$ is by definition $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ (if it exists) where $\mathcal{I}$ is the category having only identities as morphisms and having the elements of $I$ as objects.