Higher Order Yoneda?

["Christopher Townsend" <cft71@hotmail.com>]

I was looking for a reference (or correction!) to the following observation
in indexed category theory.

Let E be a cartesian category and H an E-indexed category (that is H is a
functor from E^op to CAT, where CAT is some background category of possibly
large categories).

Then, if C is an internal category in E we have a categorical equivalence

Nat[Cat(_,C),H]=H(C_0)

where C_0 is the object of objects of C. The objects of Nat[Cat(_,C),H] are
the natural transformations and the morphisms are the modifications (see,
e.g. definition B1.2.1(c) in Johnstone's Elephant).

On objects, this equivalence is just Yoneda's lemma, so surely it has been
observed already that it extends to this 2-categorical statement?

Best wishes, Christopher Townsend