  We generalize and greatly simplify the approach of Lydakis and
Dundas-R\"ondigs-{\O}stv{\ae}r 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=S^1 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.
