Let $\mathcal{D}$ be a category.
Ittay Weiss wrote about Free($\mathcal{D}$) in chat with me.
He said Free($\mathcal{D}$) is the free category on the underlying graph of $\mathcal{D}$.
Is Free($\mathcal{D}$) different from $\mathcal{D}$?
I would like to know the exact definition of Free($\mathcal{D}$) and applications of the notion.
EDIT(Jan. 14, 2013)
Is Free($\mathcal{D}$) isomorphic or equivalent to $\mathcal{D}$?
Counterexamples?
Best Answer
I didn't notice until now that this isn't a new question, but since Makoto asked for a fleshed-out example, I may as well post this.
The category on one object with only the identity morphism has as underlying graph a single vertex with a loop. The free category on the single vertex with a loop is the category on one object with countably many non-identity arrows $a,a\circ a,...$ i.e. as a monoid it is $\mathbb{N}$. The counit functor, of course, just sends all these arrows to the identity. The possibly-subtle point is that these non-identity arrows $a$ are generated by an arrow that was the identity before we applied the forgetful functor. But this is necessary because not every node in every graph has a distinguished loop to map to the identity in generating a free category.