  We give the explicit construction of the product of an arbitrary family of
coalgebras, bialgebras and Hopf algebras: it turns out that the product of an
arbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of
a family of coalgebras (resp. bialgebras, Hopf algebras). The equalizers of two
morphisms of coalgebras (resp. bialgebras, Hopf algebras) are also described
explicitly. As a consequence the categories of coalgebras, bialgebras and Hopf
algebras are shown to be complete and a explicit description for limits in the
above categories is given.
