If given an abelian category $\mathcal{A}$, we can consider the bounded derived category $D^b(\mathcal{A})$. For two objects $A,B \in \mathcal{A}$, we know that there is a natural identification between $$\text{Ext}_{\mathcal{A}}^i(A,B)$$ and $$\text{Hom}_{D^b(\mathcal{A})}(A,B[i])$$ using Yoneda extensions. This is proven in Verdier's thesis (see also group of Yoneda extensions and the EXT groups defined via derived category).
I wondered whether one in general can identify the morphisms in the derived category with Yoneda extensions in the following way:
Given two complexes $E^\bullet, F^\bullet \in D^b(\mathcal{A})$, can we identify the set of morphisms $$\text{Hom}_{D^b(\mathcal{A})}(E^\bullet, F^\bullet[i])$$ with sequences of morphisms in the derived category $D^b(\mathcal{A})$ $$F^\bullet \to Z_{i-1}^\bullet \to \dots \to Z_0^\bullet \to E^\bullet $$ such that the above sequence breaks into distinguished triangles, i.e. there exist distinguished triangles $$F^\bullet \to Z_{i-1}^\bullet \to G_{i-1}^\bullet, \quad G_{i-1}^\bullet \to Z_{i-2}^\bullet \to G_{i-2}^\bullet, \dots$$ In the case that $E^\bullet,F^\bullet$ are complexes concentrated in degree 0 (i.e. objects in $\mathcal{A}$), this is precisely the Yoneda extension with the $Z_j^\bullet, G_j^\bullet$ also complexes concentrated in degree 0.
For $i=1$ and $E^\bullet, F^\bullet$ arbitrary, this also holds true since a morphism $E^\bullet \to F^\bullet[1]$ can be completed to a distinguished triangle $$G^\bullet \to E^\bullet \to F^\bullet[1]$$ which we can rotate to obtain $$F^\bullet \to G^\bullet \to E^\bullet.$$ I am curious about the general case.
