We introduce weak topological functors and show that they lift and preserve weak limits and weak colimits. We also show that if A→B is a topological functor and J is a category, then the induced functor A(superscript J)→B(superscript J) is topological. These results are applied to a generalization of Wyler's top categories and in particular to functor categories of fuzzy maps, fuzzy relations, fuzzy topological spaces and fuzzy measurable spaces.