Grothendieck construction

Summary

A functor $F \colon \mathcal{C} \to \mathbf{Cat}$ assigns a category to each object of $\mathcal{C}$. The Grothendieck construction turns $F$ into a single category $\int F$ with a projection functor to $\mathcal{C}$. It glues the categories $F(c)$ together along the functors $F(f)$. It is the bridge between functors into $\mathbf{Cat}$ and categories fibred over $\mathcal{C}$. It contains the category of elements of a set-valued functor as a special case.