## 4.4 Products of pairs

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 $p\in \mathop{Mor}\nolimits _{\mathcal C}(x \times y, x)$ and $q\in \mathop{Mor}\nolimits _{\mathcal C}(x \times y, y)$ such that the following universal property holds: for any $w\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and morphisms $\alpha \in \mathop{Mor}\nolimits _{\mathcal C}(w, x)$ and $\beta \in \mathop{Mor}\nolimits _\mathcal {C}(w, y)$ there is a unique $\gamma \in \mathop{Mor}\nolimits _{\mathcal C}(w, x \times y)$ 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.

If a product exists it is unique up to unique isomorphism. This follows from the Yoneda lemma as the definition requires $x \times y$ to be an object of $\mathcal{C}$ such that

\[ h_{x \times y}(w) = h_ x(w) \times h_ y(w) \]

functorially in $w$. In other words the product $x \times y$ is an object representing the functor $w \mapsto h_ x(w) \times h_ y(w)$.

Definition 4.4.2. We say the category $\mathcal{C}$ *has products of pairs of objects* if a product $x \times y$ exists for any $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

We use this terminology to distinguish this notion from the notion of “having products” or “having finite products” which usually means something else (in particular it always implies there exists a final object).

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