Quantum group coproducts and universality under scalar extensions

We characterize the families of bialgebras or Hopf algebras over fields for which the product in the corresponding category is finite-dimensional, answering a question of M. Lorenz: if the ground field is infinite then bialgebra or Hopf products are finite-dimensional precisely when the factors are, with at most one of dimension $>1$; over finite fields the necessary and sufficient condition is instead that factors be finite-dimensional with at most finitely many of dimension $>1$; finally, these statements hold for coalgebras as well, provided the family is finite. We also characterize (a) finite field extensions as precisely those whose underlying scalar extension functor preserves coalgebra, or bialgebra, or Hopf algebra products (correcting an error in the literature); (b) algebraic field extensions as those along which finite coalgebra (bialgebra, Hopf algebra) products are preserved; and (c) again algebraic field extensions as precisely those which intertwine cofree coalgebras on vector spaces, or cofree bialgebras (Hopf algebras) on algebras.