In this talk, we compare the classifying space functor from the category of small categories to the category of topological spaces to the composition of the three functors namely the inclusion of the category of small categories in the category of simplicially enriched categories, a cofibrant replacement functor, and the geometric realization of a bar construction. To make this comparison we introduce the notion of a twisted natural transformation. This is a joint work in progress with Özgün Ünlü.