Lemma 106.16.4. Let $B \to C$ be a ring map. If
the coprojections $C \to C \otimes _ B C$ are flat and
$B \to C$ is universally injective,
then $B \to C$ is faithfully flat.
Proof. The map $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(B)$ is surjective as $B \to C$ is universally injective. Thus it suffices to show that $B \to C$ is flat which follows from Descent, Theorem 35.4.25. $\square$
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