We generalize and greatly simplify the approach of Lydakis and
Dundas-Röndigs-Østvær to construct an L-stable model
structure for small functors from a closed symmetric monoidal model category V
to a V-model category M, where L is a small cofibrant object of V. For the
special case V=M=S* pointed simplicial sets and L=S1 this
is the classical case of linear functors and has been described as the first
stage of the Goodwillie tower of a homotopy functor.
We show that our various model structures are compatible with a closed
symmetric monoidal product on small functors. We compare them with other
L-stabilizations described by Hovey, Jardine and others. This gives a
particularly easy construction of the classical and the motivic stable homotopy
category with the correct smash product. We establish the monoid axiom under
certain conditions.