Todd answered your question exactly. I'll add that the fact also hold in any closed category http://ncatlab.org/nlab/show/closed+category.

One can define a monoid in a closed category as an object $R$ equipped with morphisms $R \to [R, R]$ and $I \to R$ satisfying three axioms. Then $[R, -]$ becomes a comonad.

The fact for closed monoidal categories then follows since every CMC is a closed category, and the concepts of monoids coincide.

Todd answered your question exactly. I'll add that the fact also holds in any closed category http://ncatlab.org/nlab/show/closed+category.

One can define a monoid in a closed category as an object $R$ equipped with morphisms $R \to [R, R]$ and $I \to R$ satisfying three axioms. Then $[R, -]$ becomes a comonad.

The fact for closed monoidal categories then follows since every CMC is a closed category, and the concepts of monoids coincide.