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.