Hmm, I need more time to think about this. Will continue tomorrow.

Thank you. So there may be multiple natural transformations from $\times$ to $\times$ and from $+$ to $+$. But what happens if I require the category to be cartesian closed? Does this change anything? After all, the category of vector spaces over real numbers doesn’t have all exponentials, right?

@Colin: The notation $\langle f_i\rangle_{i \in I}$ looks nice and sensible indeed.

By the way, how would you call the corresponding operaton $[\cdot, \cdot]$ for coproducts then?

Is this construction of exponentials in functor categories mentioned somewhere in the literature?

My key problem was that I didn’t know about ends and thus didn’t use them in my construction. In Section IX.5 of Categories for the Working Mathematician, Mac Lane derives a functor $S^\S : C^\S \to X$ from a functor $S : C^{\mathrm{op}} \times C \to X$ and shows that the limit of $S^\S$ is isomorphic to the end of $S$. I actually considered this $S^\S$ construction myself, but then turned to the functor $M : \mathrm{Tw}(C) \to X$. $C^\S$ is actually a subcategory of $\mathrm{Tw}(C)$, and $M$ restricted to $C^\S$ is $S^\S$. An $S^\S$ for my particular case should have the same limits as $M$.

In the context I’m working in, $\mathcal D$ is actually a cartesian closed category, so it has products. That said, I wanted to keep the approach to constructing exponentials in functor categories as general as possible. My approach requires that the functor $L$ has a limit. Maybe this implies that $\mathcal D$ has products, but I’m not sure at all. My approach became so complex, because I didn’t know about ends and thus didn’t use them.