TITLE: Quotient of a category by a free group action
QUESTION [6 upvotes]: Let Cat denote the 1-category of small categories.  The functor Mor : Cat -> Set which assigns to a category its set of morphisms (aka Hom([• -> •], -)) does not commute with most colimits.  Does it commute with quotients by free group actions?  In other words, if C is a small category and G is a group acting on C such that the action of G on the objects of C is free, does Mor(C/G) = (Mor C)/G?

REPLY [2 votes]: This came up in something I'm working on, and this question was the first hit on google.
It seems to me that this has nothing to do with G acting freely, and that the answer is always yes.  One can always define a quotient category C/G with objects (Ob C)/G and morphisms (Mor C)/G, with composition and identities defined in a more-or-less obvious manner (use the action of G to match up two arrows $f: C\to D$ and $h: gD\to E$ whose domains are equal in C/G, and then compose; note that $h\circ g(f)$ and $g^{-1} (h) \circ f$ are equivalent morphisms in the quotient).  I was a little surprised that this makes sense, but it seems to.  
I think it is elementary to check that C/G satisfies the universal property defining the colimit of the functor $BG \to$ Cat.  
Am I missing something?  Edit: YES!  See the comments below.  If you care to see what I was actually thinking about (a version of this which is actually correct) look at my answer to  this subsequent question.