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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]], and let $f \\colon Z \\to X$ and $g \\colon Z \\to Y$ be [[morphisms|859F0TFVA3]] of $\\mathcal{C}$.\nA *pushout* of $f$ and $g$ is an object $P$ with morphisms $i_1 \\colon X \\to P$ and $i_2 \\colon Y \\to P$ such that:\n\n1. $i_1 \\circ f = i_2 \\circ g$.\n2. For every object $Q$ and morphisms $j_1 \\colon X \\to Q$ and $j_2 \\colon Y \\to Q$ with $j_1 \\circ f = j_2 \\circ g$, there is a unique morphism $u \\colon P \\to Q$ with $u \\circ i_1 = j_1$ and $u \\circ i_2 = j_2$.\n\n## Remarks\n\nEquivalently, $(P, i_1, i_2)$ is a [[colimit|BJS1CGTKQ5]] of the diagram $X \\xleftarrow{f} Z \\xrightarrow{g} Y$.\nA pushout is unique up to a unique isomorphism, and one writes $P = X \\sqcup_Z Y$ or $X +_Z Y$.\n\nExample: in the category of sets, let $\\sim$ be the smallest equivalence relation on $X \\sqcup Y$ with $f(z) \\sim g(z)$ for all $z \\in Z$.\nLet $P = (X \\sqcup Y)/{\\sim}$, and let $i_1, i_2$ send an element to its class.\nClause 1 holds by the choice of the relation.\nFor clause 2, the map $u$ must send the class of $x$ to $j_1(x)$ and the class of $y$ to $j_2(y)$; this is well defined because $j_1 \\circ f = j_2 \\circ g$.\n\nNon-example: take $P = X \\sqcup Y$ with the two inclusions and no identification.\nIf $Z$ is not empty, then $i_1(f(z))$ and $i_2(g(z))$ lie in different parts, so clause 1 fails.\n\nIn the category of groups the pushout is the amalgamated free product $X *_Z Y$.\n",
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