In chapter 4 of "Galois Theories" by Borceux and Janelidze, we introduced the notion of Galois descent morphism
Definition Let $\mathcal{C}$ be a category with pullbacks. A morphism $f:X \to Y$ is an effective descent morphism when the functor "pullback along $f$" $$f^\ast : \mathcal{C}/Y \to \mathcal{C}/X$$is monadic.
Definitioin Let $ \sigma: R \to S$ be a morphism of rings. Write $\beta$ for the unit of the adjunction$$\mathrm{Sp}_S:S\text{-}\mathbf{Alg}^\mathrm{op } \to \mathbf{Prof}/\mathrm{Sp}(S);A \mapsto(\mathrm{Sp}(A)\to\mathrm{Sp}(S))\\ \mathcal{C}_S:\mathbf{Prof}/\mathrm{Sp}(S)\to S\text{-}\mathbf{Alg}^\mathrm{op };(X,f)\mapsto \mathrm{Hom}((X,f),(\coprod_{M:\text{maximal regular ideal}}S/M,p))$$An $R$-algebra $A$ is split by $\sigma$ when the morphism $$\beta_{S\otimes_R A}:\mathcal{C}_S\mathrm{Sp}_S(S\otimes_R A)\to S\otimes_R A$$is an isomorphism
Definition A morphism $\sigma: R\to S$is of rings is of Galois descent when
(i)$\sigma$ is an effective descent morphism in $\mathbf{Ring}^\mathrm{op}$,
(ii)for every object $(X,\varphi)\in\mathbf{Prof}/\mathrm{Sp}(S)$, the $R$-algebra $\mathcal{C}_S(X,\varphi)$ is split by $\sigma$.
My question is the following: is there any relation between the notion of Galois descent defined above and the proposition usually mentioned as "Galois descent of vector space or algebra"?