Is the category of small categories locally presentable? I was wondering whether the various model structures on the category of small categories are combinatorial. I think that the ones I know are at least cofibrantly generated. In order to be combinatorial, the category of small categories must be locally presentable. Is this true or false? 
Intuitively, every small category should be $\lambda$-small, for $\lambda$ an upper bound of the set of objects and all morphism sets. This leads me to think that the answer is 'yes', but I'm not sure since I don't really know how are colimits in the category of small categories.
PS If your answer has a nice description of how to compute colimits in the category of small categories (or just push-outs and filtered colimits) I will appreciate it very much.
 A: Yes.  The theory of categories is defined by a finite limit sketch, i.e. an essentially algebraic theory, and the category of models of any finite limit sketch is locally finitely presentable.  See, for instance, section 3.D of Adamek & Rosicky.
A: As Mike says, it's locally finitely presentable because $Cat$ is (equivalent to) the category of models of a finite limit sketch. A quick way of describing this is to say that a category is (or is the nerve of) a simplicial set such that certain squares in the combinatorial definition of simplicial set (via faces, degeneracies) have to be pullbacks, as stated more precisely in the sixth proposition of this section of the nLab article on nerves. This clearly gives the notion of category in terms of a finite limit sketch. (In fact, this will work with truncated simplicial sets $C$, involving only $C_j$ up to dimension $j = 3$ if I'm not mistaken.) 
In other language, every category is a filtered colimit of finitely presented categories (meaning categories presented as coequalizers of pairs 
$$R \stackrel{\to}{\to} S$$ 
where $R, S$ are free categories on finite graphs). Colimits in $Cat$ can be a little bit unpleasant; coproducts are of course unproblematic, but coequalizers are another matter. An explicit construction of coequalizers can be found in this paper. (I mean to come back to this answer and improve it if possible.) 
A: About the construction of coequalizer in $Cat$: 
I call a graph unitary if for each objet $X$ is assigned the unit arrow $1_X: X\to X$) and unitary-graph morphism is a units-preserving morphims.
Let $F, G: A\to B$ two functors, consider these as morphisms of unitary graphs, its (unitary graphs) coker $B\to C$  has for object (resp. arrows) the coker of the objects (arrows) maps: $|F|_0, |G|_0$ (resp.  $|F|_1, |G|_1$). 
Let $F(C)$ the free category on $C$ (we can take it with the same units of $C$), and $Q:=C/\sim$ the quotient of $F(C)$ with the congruence generated by $g\cdot f\sim g\circ f$ (where "$\circ$" is the composition of $B$ and "$\cdot$" the free composition of $C$), then the composition of graph morphisms $q: B\to C\to F(C)\to Q$ is also a funtor $q: B\to Q$, and represent the coker of $F, G$.
I seem it work well (take with caution).
