The usual "topological fundamental group" $\pi_1 (X,x)$ of a pointed topological space $(X,x)$ is functorial in the sense that a pointed continuous map $f\colon (X,x)\rightarrow (Y,y)$ induces a homomorphism $f_* \colon \pi_1 (X,x)\rightarrow \pi_1 (Y,y)$ via composition with $f$ i.e. $[\alpha]\mapsto [f\circ\alpha]$ where $\alpha$ is a loop in $X$ based at $x$.
I know that the étale fundamental group of a variety over $K$ with a distinguished $K$-point is supposed to be functorial in the same way (i.e. gives a functor from the category of pointed varieties over $K$ to the category of groups) but there are two technical issues I need to settle. Here is my definition of the étale fundamental group:
Let $X$ be a variety over $K$, let $x\in X(K)$ and let $\textbf{Cov}(X,x)$ denote the category of finite pointed étale covers $(Y,y)\rightarrow (X,x)$ where $y\in Y(K)$. Let $\Phi_x\colon \textbf{Cov}(X, x)\rightarrow \textbf{Set}$ denote the fibre functor sending a covering $\phi: (Y,y)\rightarrow (X,x)$ to the fibre $\phi^{-1}(x)\subseteq Y$. Then the étale fundamental group of $(X,x)$ is defined as $\pi_1^{\text{ét}} (X,x) = \operatorname{Aut}(\Phi_x)$.
Now I want to show that if $f\colon (X,x)\rightarrow (Y,y)$ is a pointed morphism of varieties, it induces a homomorphism $f_* \colon \pi_1^{\text{ét}} (X,x)\rightarrow \pi_1^{\text{ét}} (Y,y)$. I read right at the start of these notes how to begin: pick a cover $\phi\colon (Y', y')\rightarrow (Y,y)$ in $\textbf{Cov}(Y,y)$. Then apparently the pullback $\phi^*\colon X\times_Y Y'\rightarrow X$ is a finite étale cover of $X$ (i.e. an object in $\textbf{Cov}(X,x)$) and the fibre of $\phi^*$ above $x$ is isomorphic to the fibre of $\phi$ above $y$.
My questions are:
- Why is $\phi^* \colon X\times_Y Y' \rightarrow X$ a finite étale cover of $X$?
- How can I show the claim that $\Phi_x (X\times_Y Y', \phi^*) \cong \Phi_y (Y', \phi)$?
I will mainly be restricting my varieties $X$ and $Y$ to be elliptic curves, so an explanation assuming this would also be very welcome.
Edit: Kevin Carlson's comments below have helped me solve question 2 by myself. I include a proof here for completeness, but because I still don't have an answer to question 1 I have not posted this as an answer.
To get a canonical isomorphism $\Phi_x (X\times_Y Y')\cong \Phi_y (Y)$, note that $\Phi_x (X\times_Y Y') = \text{Spec}(K)\times_X (X\times_Y Y')$ and $\Phi_y (Y') = \text{Spec}(K)\times_Y Y'$ where $X\xleftarrow{x}\text{Spec}(K)\xrightarrow{y} Y$ are the $K$-points. Then since $f\circ x = y$, we have a commutative diagram where the two inner squares are pullbacks:
Then from the pullback pasting lemma (see the section "pasting of pullbacks") the outer rectangle is a pullback diagram and so it follows that $\Phi_x (X\times_Y Y')$ has the UMP of the fibre product $\text{Spec}(K)\times_Y Y' = \Phi_y (Y)$, so must be isomorphic.
