The context isn't a property of the monad abstraction, but of a concrete monad, like the State monad.

If you desugar everything from the State monad, than you can see, that the state 'c' is an explicit argument, which is hidden in the "sugared" version.

    import Control.Monad.State.Lazy
    
    type Counter = State Int
    
    -- using the State monad and the do notation
    increment :: Counter ()
    increment = do
       c <- get
       put $ c + 1
    
    -- using the State monad and desugar the do notation
    increment_ :: Counter ()
    increment_ = get >>= \c -> put $ c + 1
    
    -- desugar 'get'
    get_ = \c -> (c, c)
    
    -- desugar 'put'
    put_ c = \_ -> ((), c)
    
    -- desugar >>=
    bind_ a b = \c -> let (result, c') = a c in (b result) c' 
    
    -- "desugar" the State monad
    increment__ = get_ `bind_` \c -> put_ $ c + 1