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  "body": "\n## Definition\n\nLet $J$ and $\\mathcal{C}$ be [[categories|A2DE66937D]] and let $D \\colon J \\to \\mathcal{C}$ be a [[diagram|HJK75HDWFE]].\nA *colimit* of $D$ is a [[cocone|7TPBQPPN97]] $(C, \\iota)$ under $D$, with $\\iota_j \\colon D(j) \\to C$, that satisfies the following condition.\nFor every cocone $(N, \\psi)$ under $D$ there is exactly one morphism $h \\colon C \\to N$ with $h \\circ \\iota_j = \\psi_j$ for every object $j$ of $J$.\nThe object $C$ is written $\\operatorname{colim} D$ or $\\varinjlim D$, and the $\\iota_j$ are its *injections*.\n\n## Remarks\n\nEquivalently, a colimit is an [[initial object|EK85VSSRMR]] of the category of cocones under $D$, so it is unique up to unique isomorphism.\nA colimit of $D$ in $\\mathcal{C}$ is the same as a [[limit|CHS5AFJGWE]] of the corresponding diagram $D^{\\mathrm{op}} \\colon J^{\\mathrm{op}} \\to \\mathcal{C}^{\\mathrm{op}}$ in the [[opposite category|DAMK8XE1FR]].\nThe colimit of the empty diagram is an initial object.\n\nExample: in the category of sets, every diagram $D$ with $J$ small has a colimit.\nLet $\\sim$ be the smallest [[equivalence relation|0G8VEZ3WTK]] on the disjoint union $\\bigsqcup_j D(j)$ that identifies $x \\in D(j)$ with $D(u)(x) \\in D(k)$ for every $u \\colon j \\to k$.\nThe colimit is the quotient $\\left(\\bigsqcup_j D(j)\\right) / {\\sim}$.\nThe injections send $x$ to its class, and a cocone $(N, \\psi)$ induces $h([x]) = \\psi_j(x)$, which is well defined by the cocone condition.\n\nNon-example: for the diagram of two parallel functions $f, g \\colon X \\rightrightarrows Y$ with $f \\neq g$, the set $Y$ with $\\iota_Y = \\id_Y$ and $\\iota_X = f$ is not a cocone, since $\\iota_Y \\circ g = g \\neq \\iota_X$; so it cannot be a colimit.\n",
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