The Stacks project

95.6 Finite étale covers

We define a category $\textit{FÉt}$ as follows:

1. An object of $\textit{FÉt}$ is a finite étale morphism $Y \to X$ of schemes (by our conventions this means a finite étale morphism in $(\mathit{Sch}/S)_{fppf}$),

2. A morphism $(b, a) : (Y \to X) \to (Y' \to X')$ of $\textit{FÉt}$ is a commutative diagram

   $$\xymatrix{ Y \ar[d] \ar[r]_ b & Y' \ar[d] \\ X \ar[r]_ a & X' }$$

   in the category of schemes.

Thus $\textit{FÉt}$ is a category and

is a functor. Note that the fibre category of $\textit{FÉt}$ over a scheme $X$ is just the category $\textit{FÉt}_ X$ studied in Fundamental Groups, Section 58.5.