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Remark 35.4.21. If $f$ is a split injection in $\text{Mod}_ R$, one can simplify the argument by splitting $f$ directly, without using $C$. Things are even simpler if $f$ is faithfully flat; in this case, the conclusion of Lemma 35.4.20 is immediate because tensoring over $R$ with $S$ preserves all equalizers.

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