Colimit of collection of finite sets

Question

Situation: Let $\mathbb{T}$ be an endofunctor on $\mathsf{Set}$, the category of sets an suppose that $\mathbb{T}$ restricts to an endofunctor on $\mathsf{FinSet}$, the category of finite sets. That is $\mathbb{T}Y$ is finite whenever $Y$ is finite. Moreover assume $\mathbb{T}$ is monotone, that is, $Y \subseteq Z$ implies $\mathbb{T}Y \subseteq \mathbb{T}Z$.

Let $X$ be any set (possibly of infinite size). Then the finite subsets of $X$ form a directed set.

Question: Is the directed colimit of the collection $\{ \mathbb{T}Y \mid Y \subseteq X \text{ finite} \}$ equal to $\mathbb{T}X$.

Answer 1

Not necessarily. In fact it does not work for the covariant powerset functor $\mathcal{P}:\mathbf{Set}\to\mathbf{Set}$; the directed colimit of the collection of powersets of finite sets will just be their union, which is the set of finite subsets of $X$ rather than the full powerset of $X$.

Answer 2

No, of course not, at least not without more assumptions on $\mathbb{T}$.

For instance take $\mathbb{T}$ to be the covariant powerset functor. Then clearly it satisfies your conditions, but for infinite $X$, the power set of $X$ is much larger than the union of the power sets of $Y$ for $Y\subset X$ finite.