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...