Covering property after extending a morphism of adelic curves

Determine whether the induced morphism of extended adelic curves \(\alpha'\colon S_{2,K_2'}\to S_{1,K_1'}\) is a covering whenever the original morphism \(\alpha\colon S_2\to S_1\) is a covering for some associated map \(I_\alpha\).

Background

The paper considers a morphism $\alpha\colon S_2\to S_1$ of adelic curves and an algebraic extension $K_1'/K_1$. Forming the composite field $K_2'$ with $K_2$ yields an induced morphism $\alpha'\colon S_{2,K_2'}\to S_{1,K_1'}$ between the corresponding extended adelic curves, and the construction is compatible with the natural extension maps in a commutative diagram.

The authors note that isomorphisms are preserved by this extension construction, but do not establish preservation of the stronger covering property. Resolving this question would clarify the functorial behavior of coverings of adelic curves under algebraic field extensions and would support the use of extended coverings in subsequent arithmetic arguments.