  To develop a constructive description of $\mathrm{Ext}$ in categories of
coherent sheaves over certain schemes, we establish a binatural isomorphism
between the $\mathrm{Ext}$-groups in Serre quotient categories
$\mathcal{A}/\mathcal{C}$ and a direct limit of $\mathrm{Ext}$-groups in the
ambient Abelian category $\mathcal{A}$. For $\mathrm{Ext}^1$ the isomorphism
follows if the thick subcategory $\mathcal{C} \subset \mathcal{A}$ is
localizing. For the higher extension groups we need further assumptions on
$\mathcal{C}$. With these categories in mind we cannot assume
$\mathcal{A}/\mathcal{C}$ to have enough projectives or injectives and
therefore use Yoneda's description of $\mathrm{Ext}$.
