The Morse complex is an $\mathrm{\infty }$-functor

We show that the Morse complex of a compact Lie monoid can be given the structure of an $f$-bialgebra, a chain-level version of bialgebras introduced in [CHM24]; and that this assignment defines an $\mathrm{\infty }$-functor. As a consequence, we obtain two other $\mathrm{\infty }$-functors mapping closed smooth manifolds to their Morse complexes with their $A_{\mathrm{\infty }}$-coalgebra structures; and closed smooth manifolds with compact Lie group actions to their Morse complexes, with a ``$u$-bimodule’’ structure (a bimodule version for $f$-bialgebras).