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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 *cone* over $D$ is an object $N$ of $\\mathcal{C}$, the *apex*, with a [[morphism|859F0TFVA3]] $\\psi_j \\colon N \\to D(j)$ for each object $j$ of $J$.\nThe morphisms must satisfy $D(u) \\circ \\psi_j = \\psi_k$ for every morphism $u \\colon j \\to k$ of $J$.\nA *cocone* under $D$ is an object $N$ together with morphisms $\\psi_j \\colon D(j) \\to N$ such that $\\psi_k \\circ D(u) = \\psi_j$ for every $u \\colon j \\to k$.\nA morphism of cones $(N, \\psi) \\to (N', \\psi')$ over $D$ is a morphism $h \\colon N \\to N'$ with $\\psi'_j \\circ h = \\psi_j$ for all $j$.\n\n## Remarks\n\nA cone with apex $N$ is the same as a [[natural transformation|KYCPCC57FT]] $\\Delta N \\Rightarrow D$, where $\\Delta N \\colon J \\to \\mathcal{C}$ is the constant functor with value $N$; the condition $D(u) \\circ \\psi_j = \\psi_k$ is its naturality square.\nCones over $D$ and their morphisms form a category, written $\\mathrm{Cone}(D)$.\n\nExample: let $D$ be the diagram $f, g \\colon X \\rightrightarrows Y$ of two parallel functions between sets.\nLet $N = \\set{x \\in X : f(x) = g(x)}$, let $\\psi_X$ be the inclusion and let $\\psi_Y = f \\circ \\psi_X$.\nThe conditions $f \\circ \\psi_X = \\psi_Y$ and $g \\circ \\psi_X = \\psi_Y$ hold because $f$ and $g$ agree on $N$, so this is a cone.\n\nNon-example: in the same diagram take $N = X$, $\\psi_X = \\id_X$ and $\\psi_Y = f$.\nIf $f \\neq g$, then $g \\circ \\psi_X = g \\neq \\psi_Y$, so the condition fails for the morphism $g$ and this is not a cone.\n",
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