Definition 76.13.1. Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. Consider commutative solid diagrams of the form

where $T$ and $T'$ are affine schemes and $i$ is a closed immersion defined by an ideal of square zero.

We say $a$ is

*formally smooth*if given any solid diagram as above there exists a dotted arrow making the diagram commute^{1}.We say $a$ is

*formally étale*if given any solid diagram as above there exists exactly one dotted arrow making the diagram commute.We say $a$ is

*formally unramified*if given any solid diagram as above there exists at most one dotted arrow making the diagram commute.

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