  The injective stabilization of the tensor product is subjected to an
iterative procedure that utilizes its bifunctor property. The limit of this
procedure, called the asymptotic stabilization of the tensor product, provides
a homological counterpart of Buchweitz's asymptotic construction of stable
cohomology. The resulting connected sequence of functors is isomorphic to
Triulzi's $J$-completion of the Tor functor. A comparison map from Vogel
homology to the asymptotic stabilization of the tensor product is constructed
and shown to be always epic. The category of finitely presented functors is
shown to be complete and (co)complete. As a consequence, the inert injective
stabilization of the tensor product with fixed variable a finitely generated
module over an artin algebra is shown to be finitely presented. A description
of its defect and all right-derived functors is given. A surprising connection
with Buchweitz cohomology based on injectives is established
