**EDIT:** _Thanks to Harry Gindi and Anonymous below for insightful comments, I've refined the definitions here._ Recall that a [formally etale morphism](https://en.wikipedia.org/wiki/Formally_%C3%A9tale_morphism) of schemes $Y \to X$ is a morphism which has the [unique right lifting property](https://ncatlab.org/nlab/show/orthogonality#in_category_theory) with respect to all nilpotent thickenings $Z \to W$. One salient feature of nilpotent thickenings is that they are [universal homeomorphisms](https://en.wikipedia.org/wiki/Universal_homeomorphism).

**Definition:** Say that a morphism of schemes $Y \to X$ is _strongly formally etale_ if it has the unique right lifting property with respect to all universal homeomorphisms $Z \to W$. That is, for every commutative square as below, there exists a unique diagonal filler $W \to Y$, as indicated, making the two triangles commute.

$$\require{AMScd} \begin{CD} Z @>>> Y \\ @VVV \nearrow @VVV\\ W @>>> X \end{CD}$$

By definition, then, if $Y \to X$ is strongly formally etale, then $Y \to X$ is formally etale. The converse presumably does not hold. However, etale morphisms have an additional finiteness condition (etale = formally etale + locally of finite presentation) which makes me hope for an affirmative answer to the first question below:

**Questions:**

 1. Let $Y \to X$ be an etale morphism. Then is $Y \to X$ strongly formally etale?

 2. Does there exist standard terminology for "strongly formally etale"?

 3. Is there a characterization of the class of morphisms which have the unique left lifting property with respect to all etale morphisms? How about (strongly) formally etale?