The Stacks project

Definition 4.4.1. Let $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. A product of $x$ and $y$ is an object $x \times y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $p\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times y, x)$ and $q\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \times y, y)$ such that the following universal property holds: for any $w\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and morphisms $\alpha \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x)$ and $\beta \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(w, y)$ there is a unique $\gamma \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(w, x \times y)$ making the diagram

commute.