In our recent work (Effectful Lenses: There and Back with Different Monads, conditionally accepted by ICFP 2025 and attached as a supplementary material), we developed a new theory to incorporate monadic effects into lenses. Our theory allows using two different monads in the forward and backward directions, so one key challenge is to describe the interaction of different effects in the two directions. Our original formalisation involves a relation with five axioms, and in this talk, we will present an alternative formalisation based on the composition of two different monads, where effects from the two monads are cleanly separated into two distinct stages. We will show that this composition structure forms a free polymonad. With this new composition structure, three of our five axioms holds naturally. We wish to give the talk in 30 minutes (the default).