This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten.

Let A be a commutative unitary ring, let $R$ be a commutative monoid, and let $α ∶ R → A$ be a morphism of monoids with A taken as a multiplicative monoid. Given an A- module U, a U-valued α-derivation on A is a function $D ∶ A → U$ satisfying

1. (Diff1) R-triviality: $D(α(r)) = 0$ for all $r ∈ R$;
2. (Diff2) Leibniz rule: $D(ab) = aD(b) + bD(a)$ for all $a, b ∈ A$. A differential $(A, α)$-module is a pair $(U , D)$, where $U$ is an $A$-module and $D$ is a $U$-valued $α$-derivation on $A$.

A differential $(A, α)$-module is a pair $(U , D)$, where $U$ is an $A$-module and $D$ is a $U$-valued $α$-derivation on $A$.

Given differential $(A, α)$-modules $(U, D)$ and $(V, E)$, a morphism of differential $(A, α)$-modules is a morphism of $A$-modules $f : U \rightarrow V$ that satisfies $E = f ◦ D$. We obtain a category of differential $(A, α)$-modules which we denote by $Φ(A,α)$.

For an A-module U , let $Der(A,α)(U) = \{D : A → U : (U, D) ∈ Ob(Φ(A,α))\}$. This is an A-module with the structure induced by U. Given A-modules U and V and a morphism f ∈ HomA(U, V), we define $Der(A,α)(f) : Der(A,α)(U) → Der(A,α)(V)$ by $Der(A,α)(f)(D) = f ◦ D$.

Lemma 5.2. The rule Der(A,α) defines a functor A-Mod → A-Mod

The lemma 5.2 is provided without a proof.

I am left to wonder how is this map on objects defined given an A-Module U, I cant see how to read the set builder style notation for objects in a way that makes sense to me. How should one read it?

To note some of the confusing bits about this section: D, E are functions, f is a morphism of A-modules their composition is a morphism of A-modules, are morphism of A-modules composed with functions again sensible morphisms of A-modules in some way? Is there a canonical way to view A as an A-module?