This is section 5.1 from, [ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten](https://arxiv.org/pdf/2106.16165.pdf).

> 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 $\rightarrow$ $A$-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, I understand how this yields another function. Are morphism of A-modules composed with functions again sensible morphisms of A-modules in some way?