  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. 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 RepG 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.
