The Morse complex is an $\infty$-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 $\infty$-functor. As a consequence, we obtain two other $\infty$-functors mapping closed smooth manifolds to their Morse complexes with their $A_\infty$-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).