I have a situation which I believe is simple enough it has a name that I cannot find:
A category $C$ and a functor $\otimes : C \times C \to C$ and a natural transformation $a: id \times \otimes \Rightarrow \otimes \times id$ and two sided unitors (which are isomorphisms) satisfying the triangle and pentagon coherence conditions.
This is some kind of a weak version of a monidal cateogry.
How such categories are called?