Is the following true? I am struggling to prove it or see that it is false.
Let $k: \operatorname{Hom}(A,B) \to \operatorname{Hom}(A,C)$ be an arrow in the category of abelian groups for objects $A,B,C$ in an abelian category $\mathcal{A}$. Then there exists some $k':B \to C$ so that $\operatorname{Hom}(k') = k$.
It is true if you have such a $k_A$ for all $A$, natural in $A$ (it's the content of the Yoneda lemma).
However for a fixed $A$ it is far from true.
For instance pick $A=C = \mathbb {Z/2Z}, B = \mathbb{Q/Z}$, then $\hom(A,B)$ and $\hom(A,C)$ are isomorphic (both to $\mathbb{Z/2Z}$), but there is no morphism $B\to C$ inducing that isomorphism (in fact there is no nontrivial morphism $B\to C$)