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

$\xymatrix{ w \ar[rrrd]^\beta \ar@{-->}[rrd]_\gamma \ar[rrdd]_\alpha & & \\ & & x \times y \ar[d]_ p \ar[r]_ q & y \\ & & x & }$

commute.

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