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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

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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 $\text{im}\, f\stackrel{k\circ i}{\to} \ker f.$

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?

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