I am trying to prove the theorem 5.12.7 in the book “Tensor categories” by Etingof, Gelaki, Nikshych and Ostrik. The statement is as follows:
The assignments
$(\mathcal{C},F) \mapsto H = \mathrm{End}(F), H \mapsto (\mathrm{Rep}(H), \mathrm{Forget})$
are mutually inverse bijections between (1) the set of monoidal equivalence classes of finite ring categories $\mathcal{C}$ over $k$ with a quasi-fiber functor and (2) the set of equivalence classes of finite dimensional quasi-bialgebras $H$ over $k$ up to twist equivalence and isomorphism.
I could prove these assignments are well-defined and $H \sim \mathrm{End}(U_H)$ where $U_H:\mathrm{Rep}(H) \rightarrow \mathrm{Vect}_k$ is forgetful functor. But, I cannot prove the following statement.
Let $\mathcal{C}$ be a finite ring category over $k$ with a quasi-fiber functor $F$ and let $\tilde{F}:\mathrm{Rep}(\mathrm{End}(F)) \rightarrow \mathrm{Vect}_k$ be a fogetful functor. There exists monoidal equivalence $G:\mathcal{C} \rightarrow \mathrm{Rep}(\mathrm{End}(F))$ such that $\tilde{F}\circ G=F$.
I found the proof of “Coend construction version” of this theorem. But, I couldn’t find the proof of above statement. Can anyone help me?