  Tambara modules are strong profunctors between monoidal categories. They've
been defined by Tambara in the context of representation theory, but quickly
found their way in applications when it was understood Tambara modules provide
a useful encoding of modular data accessors known as mixed optics. To suit the
needs of these applications, Tambara theory has been extended to profunctors
between categories receiving an action of a monoidal category. Motivated by the
generalization of optics to dependently-typed contexts, we sketch a further
extension of the theory of Tambara modules in the setting of actions of double
categories (thus doubly indexed categories), by defining them as horizontal
natural transformations. The theorems and constructions in Pastro-Street theory
relevant to profunctor representation theorem for mixed optics are reobtained
in this context. This reproduces the definition of dependent optics recently
put forward by Vertechi and Milewski, and hinted at by previous work of the
author and his collaborators.
