  We prove that the category of cocommutative bialgebras in any symmetric
monoidal category (that has equalizers) is an S-protomodular category with
respect to a particular class of split extensions of cocommutative bialgebras.
We also obtain the ``partial'' well-known Smith is Huq condition, meaning that
two S-equivalence relations centralize each other as soon as the normal
subobjects associated with them commute in the sense of Huq.
