Let $F:\mathcal{C}\to\mathcal{D}$ be an functor between abelian categories, and $f:X\to Y$ be a morphism in $\mathcal{C}$. Suppose that $F$ maps left exact sequences to left exact sequences. I want to show that if $k:K\to X$ is a kernel for $f$, then $F(k):F(K)\to F(X)$ is a kernel for $F(f)$.
It is clear that $F(f)F(k)=0$. Now let $P$ be an object in $\mathcal{D}$ and $p:P\to F(X)$ be a morphism, with $F(f)p=0$. I need to show that there exists a unique morphism $\theta:P\to F(K)$ with $F(k)\theta=p$, and then $F(k)$ will be a kernel for $F(f)$.
I'm sure I'm missing something... where can I go from here?