The Stacks project

Is it proven somewhere in the Stacks Project that this definition for $\operatorname{Ext}^i(A,B)$, $A,B\in\mathcal{A}$ indeed yields the $i$-th derived functor of $\operatorname{Hom}$? I cannot find it neither in 13.27 nor with the searcher.

Right after 13.27.1 it is commented that we have the expected long exact sequence, but why should this $\delta$-functor be universal?

@#8785 Okay, I just realized: if $\mathcal{A}$ has enough injectives, then the $\delta$-functor $(\operatorname{Ext}_\mathcal{A}^i(A,-))_{i\geq 0}$ is universal because it is erasable, i.e., it satisfies the hypothesis of 12.12.4: For an object $B\in\mathcal{A}$, take an injection $B\to I$ into an injective object. Then $\operatorname{Ext}_\mathcal{A}^i(A,I)$ vanishes for all $i>0$ by 13.18.8. Dually, if $\mathcal{A}$ has enough projectives, one can apply the analogous argument to $(\operatorname{Ext}_\mathcal{A}^i(-,B))_{i\geq 0}$ to deduce erasibility (and thus universality).

Is the statement of this result to be found anywhere in the SP?

(Please, erase #8828, I posted it in the wrong tag.)

@#8785 Provided $\mathcal{A}$ has enough injectives, this is immediate from Lemma 13.27.2 and the definition of a derived functor. In general the $i$th derived functors may not be defined, but the Ext's are always defined.

For the interested reader, even in the absence of injectives, one can still get away with the formula

$$\operatorname{Ext}^k_\mathcal{C}(X,Y)\cong H^k(R\operatorname{Hom}_{\mathcal{C}}(X,Y))$$ provided that $R\operatorname{Hom}_{\mathcal{C}}(-,-)$ exists. See Kashiwara, Schapira, Categories and Sheaves, Theorem 13.4.1 and equation (13.4.2).