Lemma 76.13.8. Let S be a scheme. Let F, G, H : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}. Let a : F \to G, b : G \to H be transformations of functors. Assume b is formally unramified.
If b \circ a is formally unramified then a is formally unramified.
If b \circ a is formally étale then a is formally étale.
If b \circ a is formally smooth then a is formally smooth.
Proof.
Let T \subset T' be a closed immersion of affine schemes defined by an ideal of square zero. Let g' : T' \to G and f : T \to F be given such that g'|_ T = a \circ f. Because b is formally unramified, there is a one to one correspondence between
\{ f' : T' \to F \mid f = f'|_ T\text{ and }a \circ f' = g'\}
and
\{ f' : T' \to F \mid f = f'|_ T\text{ and }b \circ a \circ f' = b \circ g'\} .
From this the lemma follows formally.
\square
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