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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]].\nA *system of zero morphisms* on $\\mathcal{C}$ is a choice of a [[morphism|859F0TFVA3]] $0_{XY} \\colon X \\to Y$ for all objects $X$ and $Y$, with the following property.\nFor all objects $W, Z$ and all morphisms $g \\colon W \\to X$ and $f \\colon Y \\to Z$:\n$$f \\circ 0_{XY} = 0_{XZ} \\quad \\text{and} \\quad 0_{XY} \\circ g = 0_{WY}.$$\nA category with such a choice is a *category with zero morphisms*, and the morphisms $0_{XY}$ are its *zero morphisms*.\n\n## Remarks\n\nA category has at most one system of zero morphisms: if $0$ and $0'$ are two, then $0_{XY} = 0'_{YY} \\circ 0_{XY} = 0'_{XY}$.\nIf $\\mathcal{C}$ has a [[zero object|AQET9YHH6F]] $0$, setting $0_{XY}$ equal to the composite of the unique morphisms $X \\to 0$ and $0 \\to Y$ gives a system of zero morphisms.\n\nExample: in the category of groups, let $0_{GH}$ send every element of $G$ to the identity $e_H$ of $H$.\nFor a [[group homomorphism|0JYTG5K9Y4]] $f \\colon H \\to K$, $f(e_H) = e_K$, so $f \\circ 0_{GH} = 0_{GK}$; and $0_{GH} \\circ g$ is again constant at $e_H$, so $0_{GH} \\circ g = 0_{G'H}$ for $g \\colon G' \\to G$.\n\nNon-example: the category of sets has no system of zero morphisms.\nThere is no function at all from a non-empty set $X$ to the empty set, so $0_{X \\emptyset}$ cannot be chosen.\n",
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