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The Stacks project

Definition 4.9.1. Let $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, $f\in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, x)$ and $g\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(y, z)$. A pushout of $f$ and $g$ is an object $x\amalg _ y z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ together with morphisms $p\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x, x\amalg _ y z)$ and $q\in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(z, x\amalg _ y z)$ making the diagram

\[ \xymatrix{ y \ar[r]_ g \ar[d]_ f & z \ar[d]^ q \\ x \ar[r]^ p & x\amalg _ y z } \]

commute, and 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 $\beta \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(z, w)$ with $\alpha \circ f = \beta \circ g$ there is a unique $\gamma \in \mathop{\mathrm{Mor}}\nolimits _{\mathcal C}(x\amalg _ y z, w)$ making the diagram

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

commute.

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