I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.

Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(X,M)$ of functions $\varphi:X\to M$ that have finite support $supp(\varphi):=\{x\in X:\varphi(x)\ne 0\}$ where $0$ is the neutral element of $M$ with respect to the commuative operation $+$.

The set $F(X,M)$ has an obvious structure of commutative monoid (actually a submonoid of the power $M^X$).

Any function $f:X\to Y$ between sets induces a monoid homomorphism $Ff:F(X,M)\to F(Y,M)$ that assigns to each $\varphi\in F(X,M)$ the function $\psi:Y\to M$, $\psi:y\mapsto \sum_{x\in f^{-1}(y)}\varphi(x)$ (the latter sum is well-defined since it contains only finitely many non-zero terms).

The construction $F(X,M)$ determines a functor $F:\mathbf{Set}\to \mathbf{Mon}$ from the category $\mathbf{Set}$ of sets to the category $\mathbf{Mon}$ of commutative monoids.

If am interested if the functor $F$ has some known reserved name.

**Remark.** For some special monoids $M$ the functor $F$ is well-known in Algebra. For example,

$\bullet$ for the group $\mathbb Z$ of integers, the monoid $F(X,\mathbb Z)$ can be identified with the free Abelian group of $X$;

$\bullet$ for the 2-element cyclic group $C_2$, the the monoid $F(X,C_2)$ can be identified with the free Boolean group of $X$.

$\bullet$ for the n-element cyclic group $C_n$, the the monoid $F(X,C_n)$ can be identified with the free Abelian group of $X$ in the variety of Abelian groups satisfying the identity $x^n=1$;

$\bullet$ for the 2-element semilattice $2=\{0,1\}$ with operation $\max$, the monoid $F(X,2)$ can be identified with the free semilattice with unit over $X$.

**Added in Edit.** I see that besides downvotes no good name for the functor $F$ was suggested. I perfectly understand that $F(X,M)$ is the direct sum of $X$ copies of $M$. But this cannot be written as a short name of the functor. Or call it "the functor of $M$-th copower" (by analogy with the "functor of $n$th power" assigning to each $X$ its power $X^n$)? My previous idea was "the functor of $M$-valued finitary functions". What is better or more appropriate? Simply, I should call it somehow in a paper. Thanks for constructive comments.