Sheafifiable homotopy model categories, II

If a Quillen model category is defined via a suitable right adjoint over a sheafifiable homotopy model category (in the sense of part I of this paper), it is sheafifiable as well; that is, it gives rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions. This is chiefly useful in dealing with homotopy theories of algebraic structures defined over diagrams of fixed shape, and unifies a large number of examples.