The paper *Facets of Descent I* by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.

I like the conceptual definition of descent data as a homotopy limit, as described in this answer, but am not sure how to translate $\mathbb E$-descent to this context. Are the sites on the category of spaces changing, or is the fibration changing?

So if $\text{Č}$ is the Čech nerve of an arrow $f:X\rightarrow Y$, and $F$ is a fibration, say $f$ is an effective descent morphism if $FY=\operatorname{holim}F(\text{Č})$.

My question is whether or not its true that $f$ is an $\mathbb E$-descent morphism iff we restrict to the subcategory of arrows in $\mathbb E$, or how to describe $\mathbb E$-descent in these terms.

What's the relationship between $\mathbb E$-descent for $\mathbb E$ the class of étale maps and descent for the étale topological site?