I have a question about following isomorphisms in Ravi Vakil's "The Rising Sea" (page 746):
Why holds $$H^{\bullet}(I^A _{\bullet}, I^B _{\bullet}) \cong H^{\bullet}(A, I^B _{\bullet}) \cong Ext^{\bullet}(A, B) $$
?
Here whe have following setting: Let $\mathcal{C}$ an abelian catogory and $A_{\bullet}$ and $B_{\bullet}$ complexes in $\mathcal{C}$. $Hom(A_{\bullet}, B_{\bullet})$ can be endowed with graduation where $Hom_n(A_{\bullet}, B_{\bullet})$ are maps between $A_{\bullet}$ and $B_{\bullet}$ shifted to the right by $n$.
The map $\delta:Hom(A_{\bullet}, B_{\bullet}) \to Hom(A_{\bullet}, B_{\bullet})$ as defined in the excerpt turns $Hom(A_{\bullet}, B_{\bullet})$ to a complex $Hom_{\bullet}(A_{\bullet}, B_{\bullet})$ because it respects $\delta ^2 =0$, therefore we can consider the cohomology groups $H^{\bullet}(A_{\bullet}, B_{\bullet})$ of it.
We have $A, B \in \mathcal{C}$, futhermore $\mathcal{C}$ has enough injective objects and $I^A _{\bullet}$ (resp. $I^B _{\bullet}$) are the injective resulutions of $A$ (resp. $B$). So
$$ 0 \to A \to I_0 ^A \to I_1 ^A \to ... $$