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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]] and let $f \\colon A \\to B$ be a [[morphism|859F0TFVA3]] of $\\mathcal{C}$.\nThe morphism $f$ is an *epimorphism* if, for every object $Y$ and all morphisms $g_1, g_2 \\colon B \\to Y$, the equality $g_1 \\circ f = g_2 \\circ f$ implies $g_1 = g_2$.\n\n## Remarks\n\nEquivalently, for every object $Y$ the map $\\Hom_{\\mathcal{C}}(B, Y) \\to \\Hom_{\\mathcal{C}}(A, Y)$, $g \\mapsto g \\circ f$, is injective.\nOne writes $f \\colon A \\twoheadrightarrow B$, or says that $f$ is *epic*.\nExample: in the category of sets, every [[surjective function|01RVCMSN5N]] $f$ is an epimorphism.\nIf $g_1 \\circ f = g_2 \\circ f$ and $b \\in B$, write $b = f(a)$; then $g_1(b) = g_1(f(a)) = g_2(f(a)) = g_2(b)$.\nNon-example: the inclusion $f \\colon \\set{0} \\to \\set{0, 1}$ is not an epimorphism of sets.\nThe identity of $\\set{0, 1}$ and the constant map with value $0$ agree after $f$ but are different.\nIn the category of rings with unit, the inclusion $\\Z \\to \\Q$ is an epimorphism that is not surjective.\nTwo [[ring homomorphisms|0VTH8KPCG2]] $g_1, g_2 \\colon \\Q \\to R$ that agree on $\\Z$ agree on $m/n$, since $g_i(m/n) = g_i(m) g_i(n)^{-1}$.\n",
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