L-stable functors

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.