  We show that either of the two reasonable choices for the category of compact
quantum groups is nice enough to allow for a plethora of universal
constructions, all obtained "by abstract nonsense" via the adjoint functor
theorem. This approach both recovers constructions which have appeared in the
literature, such as the quantum Bohr compactification of a locally compact
semigroup, and provides new ones, such as the coproduct of a family of compact
quantum groups, and the compact quantum group freely generated by a locally
compact quantum space. In addition, we characterize epimorphisms and
monomorphisms in the category of compact quantum groups.
