S-protomodularity of the category of cocommutative bialgebras

We prove that the category of cocommutative bialgebras in any locally presentable symmetric monoidal is an [Formula: see text]-protomodular with respect to a particular class split extensions bialgebras. also obtain “partial” well-known Smith Huq condition, meaning two effective text]-equivalence relations centralize each other as soon normal subobjects associated them commute sense Huq.