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I'm interested in a class of endofunctors on cartesian closed categories with a quite natural definition, and am wondering whether/where this class has been studied so far (and how it is called).

Fix a cartesian closed category with exponential $\mathcal{C}(-,-)$. For any morphism $f : \mathbf{X} \to \mathbf{Y}$, let $\lambda_f : \mathbf{1} \to \mathcal{C}(\mathbf{X},\mathbf{Y})$ be the associated point.

Call an endofunctor $d$ "internal" (for lack of a better name), if for any two objects $\mathbf{X}$, $\mathbf{Y}$ there is a morphism $D : \mathcal{C}(\mathbf{X}, \mathbf{Y}) \to \mathcal{C}(d\mathbf{X}, d\mathbf{Y})$ such that for any morphism $f : \mathbf{X} \to \mathbf{Y}$ we find $\lambda_{df} = D \circ \lambda_f$.

Informally, the action of the endofunctor on the homsets is mirrored by the action of some morphism between the associated function spaces.

Have you encoutered this definition before?

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up vote 3 down vote accepted

If you also require the obvious compatibilities, you get an enriched functor.

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If my understanding is correct, then any cartesian closed category may considered as enriched over itself, so the categories $C$, $D$, $V$ in that definition would all be the same category in my case. The "obvious compatibilities" would be the requirement that all the choices of morphism $D$ work together as expected, rather than restricting the endofunctor? –  Arno Mar 2 at 2:03

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