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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 a *monomorphism* if, for every object $X$ and all morphisms $g_1, g_2 \\colon X \\to A$, the equality $f \\circ g_1 = f \\circ g_2$ implies $g_1 = g_2$.\n\n## Remarks\n\nEquivalently, for every object $X$ the map $\\Hom_{\\mathcal{C}}(X, A) \\to \\Hom_{\\mathcal{C}}(X, B)$, $g \\mapsto f \\circ g$, is injective.\nOne writes $f \\colon A \\rightarrowtail B$, or says that $f$ is *monic*.\nExample: in the category of sets, every [[injective function|09NWETVTNY]] $f$ is a monomorphism.\nIf $f \\circ g_1 = f \\circ g_2$, then $f(g_1(x)) = f(g_2(x))$ for each $x \\in X$, so $g_1(x) = g_2(x)$ by injectivity.\nNon-example: the map $f \\colon \\set{1, 2} \\to \\set{0}$ is not a monomorphism.\nThe two maps $g_1, g_2 \\colon \\set{0} \\to \\set{1, 2}$ with $g_1(0) = 1$ and $g_2(0) = 2$ satisfy $f \\circ g_1 = f \\circ g_2$ but $g_1 \\ne g_2$.\nEvery isomorphism is a monomorphism, and a composite of monomorphisms is a monomorphism.\nIf $g \\circ f$ is a monomorphism, then $f$ is a monomorphism.\n",
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