Situation 15.7.1. Let $A, A', B, B', I$ be as in Situation 15.6.1. Let $B' \to D'$ be a ring map. Set $D = D' \otimes _{B'} B$, $C' = D' \otimes _{B'} A'$, and $C = D' \otimes _{B'} A$. This leads to a big commutative diagram

of rings. Observe that we do **not** assume that the map $D' \to D \times _ C C'$ is an isomorphism^{1}. In this situation we have the functor

analogous to (15.6.3.1). Note that $L' \otimes _{D'} D = L \otimes _{D'} (D' \otimes _{B'} B) = L \otimes _{B'} B$ and similarly $L' \otimes _{D'} C' = L \otimes _{D'} (D' \otimes _{B'} A') = L \otimes _{B'} A'$ hence the diagram

is commutative. In the following we will write $(N, M', \varphi )$ for an object of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$, i.e., $N$ is a $D$-module, $M'$ is an $C'$-module and $\varphi : N \otimes _ B A \to M' \otimes _{A'} A$ is an isomorphism of $C$-modules. However, it is often more convenient think of $\varphi $ as a $D$-linear map $\varphi : N \to M'/IM'$ which induces an isomorphism $N \otimes _ B A \to M' \otimes _{A'} A = M'/IM'$.

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