The free-monoid monad as a monoid object in the endofunctor category

Let $\mathcal{S}$ be the full subcategory of $\mathbf{S}\mathbf{e}\mathbf{t}$ on the three sets $\mathrm{\varnothing }$, $1=\{\ast \}$, and $\mathbb{N}$. This category is small. The usual free-monoid construction sends

$\mathrm{\varnothing }⟼\{\epsilon \}\cong 1,1⟼1^{\ast }\cong \mathbb{N},\mathbb{N}⟼{\mathbb{N}}^{\ast }\cong \mathbb{N},$

where the first bijection sends the empty word to $\ast$, the second sends a word on one letter to its length, and the third is any fixed bijection. Denote these chosen bijections by $b_X:X^{\ast }\to T(X)$ for $X\in \mathcal{S}$. Define $T(f):=b_Y\circ f^{\ast }\circ b_X^{-1}:T(X)⟶T(Y)$ for $f:X\to Y$. This makes $T:\mathcal{S}\to \mathcal{S}$ an endofunctor. Its transported unit has component ${\eta }_X^{\mathrm{\prime }}=b_X\circ {\eta }_X:X⟶T(X),$ and its transported multiplication has component ${\mu }_X^{\mathrm{\prime }}=b_X\circ {\mu }_X\circ (b_X^{-1}{)}^{\ast }\circ b_{T(X)}^{-1}:T^2(X)⟶T(X)\mathrm{.}$ Thus these maps correspond to one-letter insertion and word concatenation under the chosen codings; they are not literally those word maps on the representative set $\mathbb{N}$.