  We study abelian envelopes for pseudo-tensor categories with the property
that every object in the envelope is a quotient of an object in the
pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor
categories for the existence of an abelian envelope satisfying this quotient
property. %We interpret extension of scalars and Deligne product of tensor
categories as abelian envelopes and study abelian envelopes for pseudo-tensor
subcategories of $\Rep G$ for an affine group scheme. $G This allows us to
interpret the extension of scalars and Deligne tensor product of tensor
categories as abelian envelopes, and to enlarge the class of tensor categories
for which all extensions of scalars and tensor products are known to remain
tensor categories. For an affine group scheme $G$, we show that pseudo-tensor
subcategories of Rep G have abelian envelopes with the quotient property, and
we study many other such examples. This leads us to conjecture that all abelian
envelopes satisfy the quotient property.
