I have read several articles which use a version of the Weil decent criterion for covers, but the reference is always to Weil's original paper (1956 - The field of definition of a variety). I would like to know how one makes the transition. Note that when I say covers I mean a morphism between covers $(f:V\rightarrow W), (f':V'\rightarrow W)$ is a $g:V\rightarrow V'$ commuting with the covering maps.
For reference, here is the theorem from Weil's paper, which I simplified assuming Galois:
Let $k/k_0$ be a finite, Galois extension, $H=Gal(k/k_0)$. Let $V$ be a projective variety defined over $k$. Elements of $H$ have a natural action on varieties and morphisms defined over $k$ (for example by acting on the coefficients if we embed into projective space). Suppose for each $\sigma, \tau \in H$, we have an isomorphism $f_{\tau,\sigma}:\sigma(V)\rightarrow \tau(V)$. Then we have a model $V_0$ over $k_0$ for $V$ if the following are satisfied:
(i) $f_{\tau,\rho}=f_{\tau,\sigma}\circ f_{\sigma,\rho}$ for all $\sigma,\tau,\rho\in H$.
(ii) $f_{\tau \omega, \sigma \omega}=\omega(f_{\tau,\sigma})$ for all $\sigma,\tau\in H$, $\omega \in Gal(k_0^{sep}/k_0)$.
Now I suspect that to make the translation, one takes $f:V\rightarrow W$ and looks at the graph $\Gamma_f\subseteq V\times W$. Let's suppose we can satisfy the criteria above (with the $f_{\tau,\sigma}$ morphisms of covers) for $\Gamma_f$. Then we get some $\Gamma_0$ defined over $k_0$ and an isomorphism $\varphi:\Gamma_0\times k \rightarrow \Gamma_f$. There are two points I can't resolve:
(1) How do we know we have $\Gamma_0\subseteq V_0\times W_0$ for some models $V_0,W_0$ of $V,W$ (we can get the models of $V$ and $W$ over $k_0$ from the covering data). And further that it is the graph of a morphism $V_0\rightarrow W_0$.
(2) How do we know that the map $\varphi$ corresponds to a morphism of covers of $W$?
Thanks
