Let $U$,$V$ be two additive categories and $F$,$G:U → V$ additive functors. If there exist natural isomorphisms in $M\in V$ and $N\in U$ $$\phi:M,N:V(M,F(N))→V(M,G(N))$$ I want to prove that there exist an isomorphism of functors $\psi:F→G$ such that $\phi_{M,N}=V(1_{M},\psi_{N})$.
I believe Yoneda lemma gives the isomorphism between $F(N)$ and $G(N)$ defining $\psi$ as $$\psi_{N}=\phi_{F(N),N}(1_{F(N)})$$ I'm having trouble mostly verifying the naturality of $\psi$.
For a morphism $u:N\to N'$ in $U$ we have$\require{AMScd}$ \begin{CD} \hom_V(F(N),F(N))@>\varphi_{F(N),N}>>\hom_V(F(N),G(N))\\ @V\hom_{V}(1_{F(N)},F(u))VV@VV\hom_{V}(1_{F(N)},G(u))V\\ \hom_V(F(N),F(N'))@>>\varphi_{F(N),N'}>\hom_V(F(N),G(N')) \end{CD} hence $\psi_NG(u)=\varphi_{F(N),N}(1_{F(N)})G(u)=\varphi_{F(N),N'}(F(u))$. On the other hand: \begin{CD} \hom_V(F(N'),F(N'))@>\varphi_{F(N'),N'}>>\hom_V(F(N'),G(N'))\\ @V\hom_{V}(F(u),F(1_{N'}))VV@VV\hom_{V}(F(u),G(1_{N'}))V\\ \hom_V(F(N),F(N'))@>>\varphi_{F(N),N'}>\hom_V(F(N),G(N')) \end{CD} hence $F(u)\psi_N=F(u)\varphi_{F(N'),N'}(1_{F(N')})=\varphi_{F(N),N'}(F(u))$.