1. An explanation of a monoid as a category with exactly one object

newtype Cat c a = Cat {getCat :: c a a}
instance Category c => Monoid (c a)

newtype Mon m a b = Mon {getMon :: m}
instance Monoid m => Category (Mon m)
--This would be more precisely written
-- data Mon m a b where
--    Mon :: m -> Mon a a
-- but that hurts performance without seeming to offer much in return

2.
newtype Op c a b = Op {getOp :: c b a}
instance Category c => Category (Op c)
instance Bifunctor c => Bifunctor (Op c)
instance Bifunctor c => Functor (Op c a)
instance Profunctor c => Contravariant (Op c a)

I see that Control.Category.Dual in the categories package offers a version of this last defined as *data* for some reason.