We know that in a closed monoidal category $V$, the two statements are equivalent:
- $T$ is a strong endofunctor
- $T$ is an (enriched) endofunctor in $V$ enriched over itself.
I'd like to know if there is a similar relationship between strong profunctors and enriched profunctors. I couldn't find the categorical definition of a strong profunctor, so I'm translating from Haskell: A strong profunctor is a functor from $C^{op}\otimes C$ to $Set$ with the following two natural transformations: $$ T(a, b) \rightarrow T(c \otimes a, c \otimes b) $$ $$ T(a, b) \rightarrow T(a \otimes c, a \otimes c) $$ An enriched profunctor would be an (enriched) functor from $C^{op}\otimes C$ to $V$. (I'm mostly interested in $V$ being $Set$.)