Suppose we have two exact sequences of $R$-modules ($R$ is a commutative ring)
$$0\rightarrow M_0\rightarrow F\rightarrow M_1\rightarrow0$$ $$0\rightarrow N_0\rightarrow G\rightarrow N_1\rightarrow0$$
such that $\text{Hom}_R(M_i, N_i)=0$, $i=1,2$. Is it possible in this case to say something about $\text{Hom}_R(F, G)$?
$\textbf{Update}$
Applying $\text{Hom}_R(-,G)$ to the first sequence we obtain the long exact sequence
$$0\rightarrow\text{Hom}_R(M_1,G)\rightarrow \text{Hom}_R(F,G)\rightarrow \text{Hom}_R(M_0,G)\rightarrow\text{Ext}_R^1(M_1,G)\rightarrow ...$$
Applying $\text{Hom}_R(M_1,-)$ to the second one I obtain
$$0\rightarrow\text{Hom}_R(M_1,N_0)\rightarrow \text{Hom}_R(M_1,G)\rightarrow \text{Hom}_R(M_1,N_1)\rightarrow\text{Ext}_R^1(M_1,N_0)\rightarrow...$$
thus $\text{Hom}_R(M_1,G)\cong \text{Hom}_R(M_1,N_0)$ and at least we have an embedding $\text{Hom}_R(M_1,N_0)\hookrightarrow\text{Hom}_R(F,G)$.