A bialgebra axiom and the Dold-Kan correspondence

We introduce a bialgebra axiom for a pair $(c,\mathrm{\ell })$ of a colax-monoidal and a lax-monoidal structures on a functor $F\text{\hspace{0.17em}⁣}:{\mathcal{M}}_1\to {\mathcal{M}}_2$ 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).