Example 8.1.10.2. Let $\operatorname{\mathcal{E}}= \operatorname{Mod}( \operatorname{ Ab })$ denote the category of pairs $(A,M)$, where $A$ is a commutative ring and $M$ is an $A$-module (see Example 5.0.0.2). Let $\operatorname{\mathcal{C}}$ denote the category of commutative rings and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be the forgetful functor $(A,M) \mapsto A$. Then (the nerve of) $U$ is a dual Beck-Chevalley fibration, in the sense of Definition 8.1.10.1. To see this, suppose we are given a commutative diagram $\sigma :$
in the category $\operatorname{\mathcal{E}}$. Then:
The morphism $f$ is $U$-cartesian if and only if the underlying map $M \rightarrow M_1$ is an isomorphism of $A$-modules.
The morphism $g'$ is $U$-cocartesian if and only if it exhibits $M_{01}$ as obtained from $M_1$ by extending scalars along the ring homomorphism $A_1 \rightarrow A_{01}$: that is, if and only if it induces an isomorphism $A_{01} \otimes _{A_1} M_1 \rightarrow M_{01}$.
The image of $\sigma $ is a pushout diagram in $\operatorname{\mathcal{C}}$ if and only if the induced map $A_0 \otimes _{A} A_1 \rightarrow A_{01}$ is an isomorphism.
If all three of these conditions are satisfied, then the composite map $A_0 \otimes _{A} M \rightarrow M_0 \rightarrow M_{01}$ is an isomorphism. Using the two-out-of-three property, we conclude that $f'$ is $U$-cartesian if and only if $g$ is $U$-cocartesian.