Lemma 15.7.5. With $A, A', B, B', C, C', D, D', I$ as in Situation 15.7.1.

Let $(N, M', \varphi )$ be an object of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \times _{\varphi , M} M'$ is flat over $B'$.

If $L'$ is a $D'$-module flat over $B'$, then $L' = (L \otimes _{D'} D) \times _{(L \otimes _{D'} C)} (L \otimes _{D'} C')$.

The category of $D'$-modules flat over $B'$ is equivalent to the categories of objects $(N, M', \varphi )$ of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$ with $N$ flat over $B$ and $M'$ flat over $A'$.

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