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Given an arrow \(f:X\to Y\) in an additive category \(\mathcal{A}\), its cokernel is its coequaliser with the zero map \(0:X\to Y\), i.e. an object \(\mathrm{coker}(f)\in\mathcal{A}_0\) together with an arrow \(\mathrm{coker}(f):Y\to\mathrm{coker}(f)\) satisfying the universal property that, for \(g:Y\to Z\) any arrow in \(\mathcal{A}\) such that \(g\circ f=0\), there is a unique arrow \(\bar{g}:\mathrm{coker}(f)\to Z\) such that \(g=\bar{g}\circ\mathrm{coker}(f)\), i.e. any map which vanishes upon precomposition with \(f\) factors through the cokernel. In the category of abelian groups, for example, the cokernel of a map is the quotient of the codomain by the image (the arrow is simply the canonical projection \(Y\to Y/\mathrm{im}f\)).