Let $R$ be a ring and $\text{Mod}\,R$ the category of $R$ modules. For two $R$-modules $X,Y$ one can define $\text{Ext}_R^n(X,Y)$ as follows. We take an injective resolution $0\rightarrow Y\rightarrow I_0 \rightarrow I_1 \rightarrow \dots,$ throw away $Y$ and apply $\text{Hom}_R(X,-)$ to obtain the cochain complex $0 \rightarrow \text{Hom}_R(X,I_0) \rightarrow \text{Hom}_R(X,I_1) \rightarrow \dots,$ denoted by $C_Y$. Then, we define $\text{Ext}_R^n(X,Y)$ as the $n$'th cohomology of $C_Y$. The horseshoe lemma shows, that a short exact sequence $0\rightarrow L \rightarrow M \rightarrow N \rightarrow 0$ of $R$-modules induces a short exact sequence of cochain complexes $0\rightarrow C_L \rightarrow C_M \rightarrow C_N \rightarrow 0,$ which induces a long exact sequence $0\rightarrow \text{Ext}^0_R(X,L) \rightarrow \text{Ext}^0_R(X,M)\rightarrow \text{Ext}^0_R(X,N)\rightarrow \text{Ext}^1_R(X,L)\rightarrow \text{Ext}^1_R(X,M)\rightarrow \text{Ext}^1_R(X,N) \rightarrow ...,$ where $\text{Ext}^0_R(X,Y) = \text{Hom}_R(X,Y)$.

Now my question is: What happens, if we do the same with a projective resolution?

That is, we take a projective resolution $\dots \rightarrow P_1\rightarrow P_0 \rightarrow Y \rightarrow 0$ throw away $Y$ and apply $\text{Hom}_R(X,-)$ to obtain the chain complex $\dots \rightarrow \text{Hom}_R(X,P_1) \rightarrow \text{Hom}_R(X,P_0) \rightarrow 0,$ denoted by $D_Y$. Then we define $F^n (Y)$ as the $n$'th homology of $D_Y$. Again the horseshoe lemma shows, that a short exact sequence $0\rightarrow L \rightarrow M \rightarrow N \rightarrow 0$ of $R$-modules induces a short exact sequence of chain complexes $0\rightarrow D_L \rightarrow D_M \rightarrow D_N \rightarrow 0$. In fact, we only need that $\text{Hom}_R(X,-)$ is an additive functor for that. The short exact sequence of chain complexes induces a long exact sequence $\dots \rightarrow F^1(L) \rightarrow F^1(M)\rightarrow F^1(N)\rightarrow F^0(L) \rightarrow F^0(M)\rightarrow F^0(N)\rightarrow 0$. However, this time we have $F^0(Y) \neq \text{Hom}_R(X,Y)$ in general. We had equality before because $\text{Hom}_R(X,-)$ is left-exact but this time we would need right-exactness. Nevertheless, one can show that this construction induces a natural transformation $F^0 \rightarrow \text{Hom}_R(X,-)$.

Now my question is: Can we describe $F^n$ better? In particular for $n=0$? Is this contrsuction useful in any way?