Given an abelian category C, we can form the Yoneda extensions $YExt^i(X,Y)$ to the equivalent classes of $i$-extensions of X by Y. Given any abelian category C, we can always formulate the derived category D(C), and define $Ext^i_C(X,Y)$ to be $Hom_{D(C)}(X,Y[i])$.

Now we can naturally associate a Yoneda $i$-extension $$0\rightarrow Y\rightarrow Z_{i-1}\rightarrow \cdots\rightarrow Z_0\rightarrow X\rightarrow 0$$ of X by Y the element $Y[i]\leftarrow[Y\rightarrow Z_{i-1}\rightarrow \cdots\rightarrow Z_0]\rightarrow X$ of $Hom_{D(C)}(X, Y[i])$.

I am wondering when this map is inj (resp. surjective, bijective)?

We know this is an isomorphism under the assumption that C admits enough injective objects or projective objects.