What's the corresponding monomorphism $Y\to X$ and epimorphism $Y\to Z_i$ which are stated in Proposition 1.2.1?

Question

In Corollary 1.2.2, for any $A$-module $X$, what's the corresponding monomorphism $Y\to X$ and epimorphism $Y\to Z_i$ which are stated in Proposition 1.2.1?

Answer 1

Take $\{Z_i\}$ to be the principal modules, i.e. $A/I$ where $I$ is an ideal. Then for every non zero module $X$, take a non zero element x in X. The module $Ax$ is principal, so that it is isomorphic to some $\{Z_i\}$, and you are done - the morphisms you take are trivial!