In Category Theory, a monoid such as $(\mathbb N, + ,0)$ can be "categorified" as a category of a single object $\mathbb N$, and morphisms as elements of $\mathbb N$. Now, consider a monad $(T:\mathbf{Set} \to \mathbf{Set},\eta, \mu)$. Such monad is a monoid in $[\mathbf{Set}, \mathbf{Set}]$, hence we can "categorify" it by defining the object to be $T \in Ob_{[\mathbf{Set}, \mathbf{Set}]}$, the morphisms would be $Hom_{[\mathbf{Set}, \mathbf{Set}]}(1,T)$, i.e. natural transformations from the terminal object in $[\mathbf{Set}, \mathbf{Set}]$, and $\mu$ would be the composition.
Now, my question is whether given our endofunctor $T$ and a fixed set $A$, can we construct a category induced by $TA$. For example, the list functor [ ] constructs free monoids over a given set, e.g. $[A]$ can be interpreted as a monoid $([A],*,[])$, where $*$ is list concatenation. And since monoids can be categorified, then $[A]$ "induces" the categorical monoid. If I'm not mistaken, [ ] can be seen as a monad. Thus, my question is if for another monad $T$, we can follow a similar procedure to get a category from $TA$.
A commutative monoid of idempotents, i.e. satisfying $a^2=a$ and $ab=ba$, correspond to partial orders with joins, i.e. strict thin finitely cocomplete categories, via $a\to b$ if $a=ab$. Moreover, monoid homomorphisms then correspond to product-preserving functors between the corresponding categories.
The free commutative monoid of idempotents on a set is its power set with binary operation given by union, and unit given by the empty set. The corresponding category/partial is that of subsets ordered by inclusion. Associating functions between sets to their image-of-subset functions determines the covariant power set functor.
This underlies a monad with multiplication given by taking the union of a set of a subsets, and unit given by the singleton function sending an element of a set to its singleton.