The Stacks project

Remark 59.65.3. The equivalences of Lemmas 59.65.1 and 59.65.2 are compatible with pullbacks. For example, suppose $f : Y \to X$ is a morphism of connected schemes. Let $\overline{y}$ be geometric point of $Y$ and set $\overline{x} = f(\overline{y})$. Then the diagram

\[ \xymatrix{ \text{finite locally constant sheaves of sets on }Y_{\acute{e}tale}\ar[r] & \text{finite }\pi _1(Y, \overline{y})\text{-sets} \\ \text{finite locally constant sheaves of sets on }X_{\acute{e}tale}\ar[r] \ar[u]_{f^{-1}} & \text{finite }\pi _1(X, \overline{x})\text{-sets} \ar[u] } \]

is commutative, where the vertical arrow on the right comes from the continuous homomorphism $\pi _1(Y, \overline{y}) \to \pi _1(X, \overline{x})$ induced by $f$. This follows immediately from the commutative diagram in Fundamental Groups, Theorem 58.6.2. A similar result holds for the other cases.

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