= Solution
The <left adjoint> to the morphism-set <functor> is the <free category on independent arrows>:
$$
L(S)=\coprod_{s\in S}[1].
$$
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 $S\to\operatorname{Mor}(\mathcal C)$.
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 $T^2S$ to the three-element set $TS$, so it cannot be invertible. \b[This adjunction is not idempotent.]
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