I presented a simple derivation of Paul’s construction of the exponential of containers. I used the opportunity to discuss the exponential of functors and the Yoneda lemma. My derivation is based on the observation that writing for the exponentiation of functors and is a Napieran functor.
This can be shown using the Yoneda lemma. Then given a container and a functor we can reason as follows:
This shows that one can exponentiate any functor with a container (predicatively) and that exponent of containers is a container, since we know that is a container and containers are closed under composition and products. Expanding the definition gives rise to Paul’s definition.