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Seems like Reader monad composed with a strong monad produces a monad, am I right?

Take a Cartesian (or monoidal) closed category; define Reader monad for a given object $E$ as $X \mapsto X^E$; and take a strong monad $M$ (strong means preserves product or tensor product). Now the ...
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Cartesian-closed categories of algebras

If the Kleisli-category of a monad is Cartesian-closed, can we say when the category of Eilenberg-Moore algebras is?