Coherence theorem for monoidal category with strong endofunctor

Question

I am interested in a reference to a Mac Lane-type coherence theorem as follows. I have a symmetric monoidal category $C$ endowed with an endofunctor $\square$ which is a strong monoidal functor, i.e. there is a natural isomorphism $m_{a, b} \colon \square a \otimes \square b \cong \square(a \otimes b)$ and a natural isomorphism $m_0\colon 1 \cong \square 1$. A large class of diagrams consisting only of associator, unitors, identities, $m$ and tensor product of these commute. I know that this kind of coherence theorems has been dealt with using very general results about algebras and pseudo-algebras of 2-monads, but I am not very familiar with this huge theory and there are some subtleties that look pretty tricky to me: https://ncatlab.org/nlab/show/2-monad#algebras_and_pseudoalgebras_over_a_2monad