Supose $\mathcal{C}$ an additive category. Let $F: \mathcal{C} \rightarrow \mathcal{C}$ and additive functor. If $k: K \rightarrow X$ is the kernel of the morphism $f: X \rightarrow Y$ and we have that exists isomorphisms $\phi: F(X) \rightarrow X$ and $\psi: F(Y) \rightarrow Y$ such that $f \circ \phi = \psi \circ F(f)$ can i say that $\phi \circ F(k): F(K) \rightarrow X$ is a kernel?

I know that will exist a morphism $\iota: F(K) \rightarrow K$ such that $\phi \circ F(k) = k \circ \iota$ but, i need that this morphism $\iota$ to be an isomorfism, and i can't see why...

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    $\begingroup$ Does this question have another context or a motivation? It seems to me that the answer is "no for trivial reasons", but that the problem, which is "hiding" at the moment, might have a positive answer. $\endgroup$ – Martin Brandenburg Dec 16 '14 at 15:45

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