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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 $$\text{im}\, f\to \ker f:$$

1. Consider the diagram where the morphisms $$i$$ and $$j$$ are obtained by definition and $$t$$ is obtained by the universal property of $$\text{coker}\, f$$. Note that

$$f\circ i=t\circ f\circ i=t\circ 0=0,$$ thereforeby universal property of $$\ker f$$, exists a morphism between $$\text{im}\, f$$ and $$\ker f$$.

1. Again, consider the sequence $$\text{im}\, f\stackrel{i}{\to} x\stackrel{j}{\to} \text{coker}\, f.$$ Since $$f\circ f$$, then by universal property of $$\ker f$$ exists a morphism $$x\stackrel{k}{\to} \ker f$$. Then $$\text{im}\, f\stackrel{k\circ i}{\to} \ker f.$$
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The text says: Since $$f\circ f\equiv 0$$, then exists a monomorphism $$\text{im}\, f\to \ker f$$. Is any of these induced morphism a monomorphism?