 Free category on independent arrows

Take two distinct objects and one nonidentity arrow for each element of $S$, without arrows between different copies. A functor from this category to $\mathcal{C}$ is exactly a choice of one arbitrary arrow of $\mathcal{C}$ for each element of $S$. Consequently this construction is left adjoint to the functor taking the set of all arrows of a small category. Its free object has three arrows per generator, including the two identity arrows.