 Solution

The left adjoint to the morphism-set functor is the free category on independent arrows:Each copy has two objects, their identities, and one nonidentity arrow from the first object to the second. A functor $L(S)\to \mathcal{C}$ is specified by one arbitrary arrow of $\mathcal{C}$ for each $s$, since that arrow specifies the images of both objects as well. This proves the natural adjunction bijection with functions $S\to \mathrm{Mor}(\mathcal{C})$.

The induced monad on sets has $T(S)=S\times \{\text{source identity},\text{generator},\text{target identity}\}$. For a singleton $S$, its multiplication is a function from the nine-element set $T^{2}S$ to the three-element set $TS$, so it cannot be invertible. This adjunction is not idempotent.