One of the simpler free monoids is the list.
I think a list category would have lists of objects as objects and its maps would also be lists.
A list of maps [a, b, c] where a: A -> A', b: B -> B', c: C -> C' would map [A, B, C] -> [A', B', C']
Recursion is a bit messy but it's all simple enough.
What would a free "locally monoidal" category look like?
IIRC a locally Cartesian category basically possesses a symmetric monoid in C/c forall c.
A "locally monoidal" category would just have a monoid in C/c.
The details of a free locally monoidal category seem kind of messy to work out. It's not quite just a list of C/c because it needs to be a list forall c.
I suppose another possibly simpler approach would be a sort of difference list.
The category of endofunctors C -> C possesses a monoid.
I don't really get a category of "natural slice endofunctors" so to speak forall x, C/x -> C/x though