Let $\textbf{CART}$ be a category where the objects are all Cartesian closed categories (henceforth shortened as CCC). Is there any way to define the arrows so that $\textbf{CART}$ itself becomes Cartesian closed? I am particularly interested in arrows that preserve the dinatural transformations of the simply typed lambda calculi in the different CCCs.
Possible start for solution [EDIT: ruled out by მამუკა ჯიბლაძე , see comment]: There is one rather obscure family of functors called Cartesian closed functors, that preserves the products and exponentials of the domain CCC. Given two CCCs $\textbf{C}$ and $\textbf{D}$, the collection of all Cartesian closed functors from $\textbf{C}$ to $\textbf{D}$ seems like a good candidate for the exponential $\textbf{D}^\textbf{C}$, but will the natural transformations between those functors make arrows in the category $\textbf{D}^\textbf{C}$ so that it becomes Cartesian closed? That is, $\textbf{D}^\textbf{C}$ is only a good exponential in $\textbf{CART}$ if it is an object in $\textbf{CART}$, but by definition of $\textbf{CART}$, it then must be a CCC.
