I know that the free functor is adjoint to the forgetful functor, which makes me doubt my hypothesis, but:
My thought is: If you start out with a diagram, and you generate a free category with it, what you are doing is: “taking all the compositions and distinguishing between the result based on how you did the composition”
But when you take the same diagram, and you generate a poset category from it, i.e. make sure that there is only one arrow for each pair of objects, what you’re doing is “taking all the compositions and ignoring the difference between how you took the composition”.
In some intuitive sense, they are opposite. Is there a formal sense that makes this precise? Is there some adjointness relation between some formalization of these concepts?