  Let $(\mathcal{A},\mathcal{E})$ be an exact category. We establish basic
results that allow one to identify sub(bi)functors of
$\operatorname{Ext}_{\mathcal{E}}(-,-)$ using additivity of numerical functions
and restriction to subcategories. We also study a small number of these new
functors over commutative local rings in details, and find a range of
applications from detecting regularity to understanding Ulrich modules.
