  We prove a number of results concerning monomorphisms, epimorphisms,
dominions and codominions in categories of coalgebras. Examples include: (a)
representation-theoretic characterizations of monomorphisms in all of these
categories that when the Hopf algebras in question are commutative specialize
back to the familiar necessary and sufficient conditions (due to Bien-Borel)
that a linear algebraic subgroup be epimorphically embedded; (b) the fact that
a morphism in the category of (cocommutative) coalgebras, (cocommutative)
bialgebras, and a host of categories of Hopf algebras has the same codominion
in any of these categories which contain it; (c) the invariance of the Hopf
algebra or bialgebra (co)dominion construction under field extension, again
mimicking the well-known corresponding algebraic-group result; (d) the fact
that surjections of coalgebras, bialgebras or Hopf algebras are regular
epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in
particular, the fact that embeddings of compact quantum groups are equalizers
in the category thereof, generalizing analogous results on (plain) compact
groups; (f) coalgebra-limit preservation results for scalar-extension functors
(e.g. extending scalars along a field extension $\Bbbk\le \Bbbk'$ is a right
adjoint on the category of $\Bbbk$-coalgebras).
