  We introduce a bialgebra axiom for a pair $(c,\ell)$ of a colax-monoidal and
a lax-monoidal structures on a functor $F\colon \mathscr{M}_1\to \mathscr{M}_2$
between two (strict) symmetric monoidal categories. This axiom can be regarded
as a weakening of the property of $F$ to be a strict symmetric monoidal
functor. We show that this axiom transforms well when passing to the adjoint
functor or to the categories of monoids. Rather unexpectedly, this axiom holds
for the Alexander-Whitney colax-monoidal and the Eilenberg-MacLane lax-monoidal
structures on the normalized chain complex functor in the Dold-Kan
correspondence. This fact, proven in Section 2, opens up a way for many
applications, which we will consider in our sequel paper(s).
