If $\mathcal{A}$ is any abelian category, then for all objects $X,Y$ in $\mathcal{A}$ and for all integers $i \geq 0$, there is an natural isomorphism $$\mathrm{Ext}_\mathcal{A}^i(X,Y) \simeq \mathrm{Hom}_{D(\mathcal{A})}(X,Y[i]),$$ where the left-hand side denotes the Yoneda $\mathrm{Ext}$-group in $\mathcal{A}$ and the right-hand side denotes the morphisms between $X$ and the shift of $Y$ by $i$ in the derived category $D(\mathcal{A})$ of $\mathcal{A}$. Note that this holds without any assumption on $\mathcal{A}$ (e.g. existence of enough injectives/projectives), cf. Verdier's thesis III.3.2.

Now, I am interested in an exact category $\mathcal{E}$ (in the sense of Quillen, i.e. a strictly full subcategory closed under taking extensions of an abelian category $\mathcal{A}$).
Exact categories encapsulate just enough of the formalism of abelian categories to define the Yoneda $\mathrm{Ext}$-groups (as sequences $E_1E_2...E_i$ of short exact sequences with compatible ends).
Moreover, under certain assumptions one can define the derived category $D(\mathcal{E})$ of $\mathcal{E}$.
In my situation, $\mathcal{E}$ has kernels so that $D(\mathcal{E})$ can be defined as in [Beilinson-Bernstein-Deligne, *Faisceaux pervers*, §1.1.4].
In this setup, do we have a natural isomorphism
$$\mathrm{Ext}_\mathcal{E}^i(X,Y) \simeq \mathrm{Hom}_{D(\mathcal{E})}(X,Y[i])$$
for all objects $X,Y$ in $\mathcal{E}$ and for all integers $i \geq 0$?