Internal natural transformations and Frobenius algebras in the Drinfeld center

For M and N finite module categories over a finite tensor category C, the category Rex_C(M,N) of right exact module functors is a finite module category over the Drinfeld center Z©. We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C-C-bimodule functors to objects of Z©, we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if M and N are exact C-modules and C is pivotal, then the Z©-module Rex_C(M,N) is exact. We compute its relative Serre functor and show that if M and N are even pivotal module categories, then Rex_C(M,N) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in Z©.