Under the category of Artin local $k$-algebras $(Art/k)$, let $\alpha: F\to G$ be a morphism of deformation functors. The morphism $\alpha$ is called smooth if for all small extensions $0\to N\to B\to A\to 0$ the canonical map $F(B)\to F(A)\times_{G(A)} G(B)$ is surjective . Now we define a deformation functor $D_{X,p}(A)$={$a\in h_X(\operatorname{Spec}A)|a|_{\operatorname{Spec}(A/m_A)}=p$}, where $h_X=Hom(-,X)$.
Then it’s said that a morphism $f:X\to Y$ of schemes is smooth at a point $p\in X$ such that $f(p)$ has residual field $k$, if and only if the corresponding morphism of functors $D_{X,p}\to D_{Y,f(p)}$ is smooth, by a so called “formal criterion for smoothness.” A smooth morphism is étale iff it’s also an isomorphism of tangent space.
I can’t find any criterion that can indicate this relation explicitly. Hope someone could help. Thanks!