categories: Re: Characterizing FinSet up to equivalence

[Peter Selinger <selinger@mathstat.uottawa.ca>]

baez@math.ucr.edu wrote:
> 
> Claim: FinSet is the free category with finite sums on one object.

I wonder what happens in the case of more than one generator. For
instance, the free category with finite sums on two objects is FinSet
x FinSet. In the case where the set of generators is discrete, it does
not make a difference if one also adds coequalizers, e.g.

 FinSet is the free category with finite colimits on one object.

What about the case where one has morphisms on the generators? From
[Mac Lane], we know:

 If D is any diagram (small category), then its free co-completion is 
 the Yoneda category Set^{D^op}.

Is this still true when inserting the word "finite"?

 If D is any diagram, then its free completion under finite colimits
 is FinSet^{D^op}?

And what happens if one drops the coequalizers? Does the free
completion of a diagram D under coproducts have a Yoneda-like
characterization?

-- Peter