Monoidal vs applicative - Haskell

The docs for Control.Invertible.Monoidal say

Invariant monoidal functor. This roughly corresponds to Applicative, [..]

What's rough about the correspondence, if anything?

I think what they're referring to is that the module's contents roughly correspond to the contents of Control.Applicative

Oh wait there's two places they talk about correspondence, whoops! Reading that part now

So you could define Applicative in terms of Monoidal I'm pretty sure.

You should try give it a go defining pure and <*> in terms of unit and >*<

I think they are actually exactly equivalent, not only "roughly correspondent", but the Applicative interface is nicer to work with, in practice.

It's basically the same thing as relation between

bind :: Monad m => m a -> (a -> m b) -> m b

and

join :: Monad m => m (m a) -> m a

One is more convenient, the other is closer to formal definition