Definition [efr-KUW7]

Let \mathcal {A}: \mathcal {C}^\mathrm {op} \to \mathsf {Cat} be a pseudofunctor. Then there is a category \int _{X \in \mathcal {C}}\mathcal {A}(X) defined as follows:

1. The objects are pairs {\bar {X}\in \mathcal {A}(X) \choose X \in \mathcal {C}}
2. The morphisms {\bar {X} \choose X} \to {\bar {Y} \choose Y} are pairs f: X \to Y, f^\#: \bar {X} \to \mathcal {A}(f)(\bar {Y}) \in \mathcal {A}(X)
3. Composition is given by the "chain rule" (f,f^\#) \circ (g,g^\#) = (fg, \mathcal {A}(g)(f^\#)g^\#)

There is an obvious forgetful functor \int _X \mathcal {A}(X) \to \mathcal {C}.

The category \int _X \mathcal {A}(X) is known as the Grothendieck construction of \mathcal {A}