## 4.5 Coproducts of pairs

Definition 4.5.1. Let $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. A *coproduct*, or *amalgamated sum* of $x$ and $y$ is an object $x \amalg y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $i \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x, x \amalg y)$ and $j \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(y, x \amalg y)$ such that the following universal property holds: for any $w \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and morphisms $\alpha \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x, w)$ and $\beta \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, w)$ there is a unique $\gamma \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x \amalg y, w)$ making the diagram

\[ \xymatrix{ & y \ar[d]^ j \ar[rrdd]^\beta \\ x \ar[r]^ i \ar[rrrd]_\alpha & x \amalg y \ar@{-->}[rrd]^\gamma \\ & & & w } \]

commute.

If a coproduct exists it is unique up to unique isomorphism. This follows from the Yoneda lemma (applied to the opposite category) as the definition requires $x \amalg y$ to be an object of $\mathcal{C}$ such that

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x \amalg y, w) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, w) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, w) \]

functorially in $w$.

Definition 4.5.2. We say the category $\mathcal{C}$ *has coproducts of pairs of objects* if a coproduct $x \amalg y$ 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 coproducts” or “having finite coproducts” which usually means something else (in particular it always implies there exists an initial object in $\mathcal{C}$).

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