  We investigate cofree coalgebras, and limits and colimits of coalgebras in
some abelian monoidal categories of interest, such as bimodules over a ring,
and modules and comodules over a bialgebra or Hopf algebra. We find concrete
generators for the categories of coalgebras in these monoidal categories, and
explicitly construct cofree coalgebras, products and limits of coalgebras in
each case. This answers an open question of A. Agore on the existence of a
cofree coring, and constructs the cofree (co)module coalgebra on a
$B$-(co)module, for a bialgebra $B$.
