 The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits.When dealing with morphisms formula_6 of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of formula_25-modules, where formula_25 is the structure sheaf of formula_3.Then the functor formula_28 is inappropriate, because in general it does not even give sheaves of formula_29-modules.