There is a well-known way to conclude the cocompleteness of a category from its completeness. Namely, if a category is complete, well-powered and has a cogenerating set, then it is cocomplete (easy corollary of the special adjoint functor theorem).
I was hoping I could use this to prove that the category $\mathbf{Grp}$ of groups is cocomplete. I would find it interesting, for completeness of $\mathbf{Grp}$ is easy to prove (the forgetful functor to the category of sets creates limits), while cocompletenes is not so easy (constructing free product is quite tedious).
But alas, a quick google search revealed that $\mathbf{Grp}$ doesn't have a cogenerator (it can be read off the only freely available page of this article). I suspect it doesn't have a cogenerating set, either.
Is there a categorical proof of the cocompleteness of $\mathbf{Grp}$?
Edit: I'd like to point out that I found out afterwards the following exercise 1 in section IX.1 of Mac Lane's CWM (I didn't expect to find this exercise in the chapter on "Special Limits"!)
Use the adjoint functor theorem to prove in one step that Grp has all small colimits.