It is unclear to me why in the distribution functor as well as in the multiset functor we require finite support. Below, I provided the definition from "Introduction to Coalgebra" by Bart Jacobs.
In several other sources, this is explained in a similar fashion, not at all. It is usually just stated that it is necessary for it to be finite, or to avoid infinte sums. As I do not have a background in math, I am assuming I must be missing something obvious.
Def: For a commutative monoid $M = (M,+,0)$ we define the multiset functor $M_M: Sets \to Sets$ on a set $X$ as $$ M_M(X) = \{ \varphi: X \to M \mid supp(\varphi) \text{ is finite} \} $$ where $supp(\varphi) = \{ x \in X \mid \varphi(x) \neq 0\}$ is called the support of $\varphi$.
On a function $f: X \to Y$ one has $M_M(f): M_M(X) \to M_M(Y)$ via $$M_M(f)(\varphi)(y) = \sum_{x \in f^{-1}(y)} \varphi(x) = \sum\{\varphi(x) \mid x \in supp(\varphi) \text{ with } f(x) = y\} $$