I am looking at a category $C$ equipped with an exponential, and a monad $T : C \to C$ equipped with a natural transformation $\nu : F \to G$ from the bifunctor $F : (X, Y) \mapsto T(Y^X)$ to the bifunctor $G : (X, Y) \mapsto T(Y)^X$. Original question: Does anyone know if such monads have been studied? Is there a name for such a monad?
I should say more about $T$ and $\nu$. My $T$ is strong with respect to the product $\times$ used to define the evaluation morphism $\varepsilon_{X,Y} : X \times Y^X \to Y$. Writing $\sigma_{X,Y} : X \times TY \to T(X \times Y)$ for the strength natural transformation, I now see that my $\nu_{X,Y} : T(Y^X) \to T(Y)^X$ is adjoint to the composite:
$$ X \times T(Y^X) \stackrel{\sigma_{X,Y^X}}{\longrightarrow} T(X \times Y^X) \stackrel{T(\varepsilon_{X, Y})}{\longrightarrow} TY $$
So the answer to the original question for that particular $\nu$ is that its existence is just a by-product of $T$ being strong. I suppose the question remains whether there is any interest in abstracting over $\nu$. The comment by Maciej Piróg below suggests that the existence of a $\nu$ satisfying suitable naturality conditions might be equivalent to the strength of $T$.