In a complete, cocomplete and well-powered category with zero object consider the canonical factorization of a morphism $f=k\circ \mathrm{Coim}(f)$. Does cocomplete+complete+well-powered guarantee that $k$ is a monomorphism?

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definitionof coimage. The coimage of $f$ is supposed to be the universal epimorphism through which $f$ factors.) – Zhen Lin Sep 23 '13 at 11:16