Theorem 53.6.2. Let $X$ be a connected scheme. Let $\overline{x}$ be a geometric point of $X$.

The fibre functor $F_{\overline{x}}$ defines an equivalence of categories

\[ \textit{FÉt}_ X \longrightarrow \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} \]Given a second geometric point $\overline{x}'$ of $X$ there exists an isomorphism $t : F_{\overline{x}} \to F_{\overline{x}'}$. This gives an isomorphism $\pi _1(X, \overline{x}) \to \pi _1(X, \overline{x}')$ compatible with the equivalences in (1). This isomorphism is independent of $t$ up to inner conjugation.

Given a morphism $f : X \to Y$ of connected schemes denote $\overline{y} = f \circ \overline{x}$. There is a canonical continuous homomorphism

\[ f_* : \pi _1(X, \overline{x}) \to \pi _1(Y, \overline{y}) \]such that the diagram

\[ \xymatrix{ \textit{FÉt}_ Y \ar[r]_{\text{base change}} \ar[d]_{F_{\overline{y}}} & \textit{FÉt}_ X \ar[d]^{F_{\overline{x}}} \\ \textit{Finite-}\pi _1(Y, \overline{y})\textit{-Sets} \ar[r]^{f_*} & \textit{Finite-}\pi _1(X, \overline{x})\textit{-Sets} } \]is commutative.

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