group of Yoneda extensions and the EXT groups defined via derived category 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.
 A: If we take $X$ and $Y$ in the abelian category $C$, and regard them as objects 
of the derived category of cochain compelexes (by placing them in degree 0), 
then $Hom_{D(C)}(X,Y[n]) = Ext^n(X,Y)$, where the left-hand side is Hom in the derived category from $X$ to a shift of $Y$, and the right-hand side is a Yoneda Ext group.
For this reason, when $A$ and $B$ are any two objects in the derived category, one 
write $Ext^n(A,B) = Hom(A,B[n]).$
Assuming that what I've written here is a faithful interpretation of your notation, then I can summarize things by saying that the answer to your question is yes.
A: For $A$ abelian artinian the higher Yoneda Ext $YExt^i(-,+)$ of objects $-,+\in A$ is the same as in  $B = Pro (A)$. Now $A$ is thick in $B$ and yields $D^b(A)$ fully faithful in $D^b(B)$ given by those complexes with homology in $A$.
Since $YExt^i_A(-,+) = Ext^i_B(-,+) = Hom_{D^b(B)}(-,+[i])  =
Hom_{D^b(A)}(-,+[i])$ we get
$YExt^i_A(-,+) = Hom_D^b(A)(-,+[i])$
for such an Artinian category $A$.
Similarly for $A$ abelian Noetherian using $Ind (A)$.
A: You can look at F. Oort "Yoneda extensions in abelian categories" in Math Annalen. 
He proves that the map is an isomorphism for artinian categories as the Yoneda Ext groups are the same in the category and in its category of pro-objects and the latter has enough projective objects. 
A: It is true in general and Verdier did the computation in his thesis Des Catégories Dérivées des Catégories Abéliennes Chap III Sect 3 ... the thesis is on line!
