So, I'm reading through some notes on the etale fundamental group (mostly Murre, but also some other notes I have), and I find it confusing how in a galois category $\mathcal{C}$ with fundamental functor $F$, the automorphism group $\text{Aut}(X)$ of an object $X\in\mathcal{C}$ can act on $F(X)$ ``on the right''. I know every left-action is equivalent to a right action by acting by the inverse, but it doesn't seem right that you should have to do that. Furthermore, given an object $X$ and an element $\xi\in F(X)$, Murre describes an injection (bijection if $X$ is galois) $\text{Aut}(X)\hookrightarrow F(X)$ by sending $u\in\text{Aut}(X)$ to $F(u)(\xi)$. This implies to me that we're not (don't need to?) do this inverse-action business.

For example, suppose $X$ is a galois object, and suppose $S$ is the terminal object for $\mathcal{C}$. Then, there is a natural (though not canonical) identification of $\text{Hom}(X,X) = \text{Aut}(X)$ with $F(X)$, and so it's natural for $\text{Aut}(X)$ to act on itself (which is identified with $F(X)$) on the right (by right-multiplication).

If $X$ is not a galois object, then we can still pick a galois object $Y$ over $X$, and again we may identify $\text{Hom}(Y,X) = F(X)$. In this case, $\text{Aut}(Y)$ naturally acts on the right on $\text{Hom}(Y,X)$ and hence on $F(X)$. However, this is not an action of $\text{Aut}(X)$. In classical galois theory, we could just restrict automorphisms of $Y$ to automorphisms of $X$, and hence get a right action of $\text{Aut}(X)$ on $F(X)$, but this restriction map doesn't seem to be available in the abstract language of galois categories.

Can someone explain how I should think of Aut$(X)$ acting on the right of $F(X)$?

thanks

- will