Trait specialization as a cover functor with geometric basepoint paths

The total space $X$ is connected. Indeed it is Noetherian and has finitely many irreducible components, so each connected component is open and closed. Each nonempty component meets $X_k$, since it is proper over the local trait and its closed nonempty image contains the closed point. A partition of $X$ would therefore partition connected $X_k$ into two nonempty open and closed sets, a contradiction. Consequently [F2] applies to the images $x_0,x_{\eta }$ of the chosen geometric basepoints as points of $X$. A path means a natural isomorphism $P:F_{x_0}\to F_{x_{\eta }}$ on $\mathrm{FEt}(X)$. Such paths exist under AC: take the product, over a small skeleton of covers, of the finite sets of bijections between their two fibres. Naturality and the identities impose closed conditions. Any finitely many such conditions are satisfied by a connected Galois cover simultaneously trivializing the covers involved; choose a point over each basepoint in that Galois cover, and use its unique mapping property in [F2] to identify the fibres of every involved cover compatibly with all maps. These conditions have the finite intersection property, so compactness of the product gives a path.

By [F1] the quasi-inverse comes with a natural isomorphism $rE\cong \mathrm{i}\mathrm{d}$. It identifies $F_{x_0}(E(V))$ with $F_{{\bar{x}}_0}(V)$. Generic pullback identifies $F_{x_{\eta }}(E(V))$ with $F_{{\bar{x}}_{\eta }}(T(V))$: both are exactly the lifts of the same geometric point to the same cover. Thus $P$ induces a natural isomorphism $\tau :F_{{\bar{x}}_0}\to F_{{\bar{x}}_{\eta }}\circ T$. For $g\in \mathrm{Aut}(F_{{\bar{x}}_{\eta }})$, define on each special cover $V$ $\mathrm{sp}(g{)}_V={\tau }_V^{-1}{g}_{T(V)}{\tau }_V\mathrm{.}$ Naturality of $g$ and $\tau$ makes this an automorphism of the special fibre functor, and componentwise composition makes $\mathrm{sp}$ a homomorphism. Its continuity follows since the action on any finite special fibre factors through the continuous action on the single finite generic cover $T(V)$; intersections of these action kernels form the topology in [F2]. This explains the direction without assuming any equivalence from special covers to all generic covers.

If ${\tau }^{\mathrm{\prime }}$ arises from a different path, put $a={\tau }^{-1}{\tau }^{\mathrm{\prime }}$, a natural automorphism of the special fibre functor. Then ${\mathrm{sp}}^{\mathrm{\prime }}(g)=a^{-1}\mathrm{sp}(g)a$ by direct substitution. Replacing the quasi-inverse gives its unique natural isomorphism compatible with $rE\cong \mathrm{i}\mathrm{d}$, since $r$ is fully faithful. Transporting $\tau$ through that isomorphism preserves the formula. For algebraically closed field extensions, [F3] identifies the cover categories, and the fibres of finite étale covers are finite sets unchanged by algebraically closed field extension. Applying these identifications to $T$ and $\tau$ gives exactly the transported homomorphism. Arbitrary geometric basepoints are handled by the same path construction of step 1.1 on the corresponding connected smooth proper fibre.

Let $R\to R^{\mathrm{\prime }}$ be local injective between complete DVRs and inducing the identity on $k$, and let $X^{\mathrm{\prime }}=X{\times }_R{R}^{\mathrm{\prime }}$. For a special cover $V$, the pullback $E(V){\times }_R{R}^{\mathrm{\prime }}$ is a cover of $X^{\mathrm{\prime }}$ restricting to $V$. By [F1] applied to $X^{\mathrm{\prime }}\mathrm{/}{R}^{\mathrm{\prime }}$, it is naturally isomorphic to $E^{\mathrm{\prime }}(V)$ for any compatible quasi-inverse $E^{\mathrm{\prime }}$. The isomorphism is uniquely determined by the special-fibre identity. Generic pullback then identifies their $T$ functors after a common algebraically closed generic overfield. Compatible paths identify their $\tau$ maps; the formula in step 1.2 gives the same specialization homomorphism. If an independently chosen path is used, step 2.1 supplies exactly the conjugacy qualification. The AC use is precisely [F4] and the compactness construction of step 1.1.