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The aim of this paper is to interpret the Grothendieck construction in the monoidal world. That is to say, we restrict the equivalence between fibred categories and pseudo functors to the case of categories having only a single object. On other way of expressing this is to say that we are given a monoid homomorphism. Though only a specialisation, we discover many pleasant results and interpret many things in a new light. We also touch upon the case of finite groups as an example.
Ilia Pirashvili (Sun,) studied this question.