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.
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