Example 5.0.0.2. Let $\operatorname{Mod}(\operatorname{ Ab })$ be the category defined above and let $\operatorname{CAlg}(\operatorname{ Ab })$ denote the category of commutative rings, so that the construction $(A,M) \mapsto A$ determines a forgetful functor $U: \operatorname{Mod}(\operatorname{ Ab }) \rightarrow \operatorname{CAlg}(\operatorname{ Ab })$. Then:
A morphism $(u,f): (A,M) \rightarrow (B,N)$ in the category $\operatorname{Mod}(\operatorname{ Ab })$ is $U$-cartesian if and only if the underlying $A$-module homomorphism $f: M \rightarrow N$ is an isomorphism (so that the $A$-module $M$ is obtained from the $B$-module $N$ by restriction of scalars along the ring homomorphism $u$).
A morphism $(u,f): (A,M) \rightarrow (B,N)$ in the category $\operatorname{Mod}(\operatorname{ Ab })$ is $U$-cocartesian if and only if the underlying $A$-module homomorphism $f: M \rightarrow N$ induces a $B$-module isomorphism $B \otimes _{A} M \simeq N$ (so that the $B$-module $N$ is obtained from the $A$-module $M$ by extension of scalars along the ring homomorphism $u$).