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Induced morphism between image and ker (category theory)

Context question: “Differential on a object of an abelian category” Let $C$ be an abelian category and $x$ an object of $C$. Consider a morphism $f: x\to x$ such that $f\circ f=0.$ I can see two ways to define a morphism $\text{im}\, f\to \ker f:$ Consider the diagram where the morphisms $i$ and $j$ are […]