I'm currently a little confused on what to show. I don't want any help on the the proof, only the structure of the proof.
My exercise is to show that for a semi simple category $\mathcal{C}$ the following holds: the structure of a k-linear category over $\mathcal{C}$ is the same as $\mathcal{C}$ being a module category over $\mathrm{vect}_k$, k a field.
As is said, I don't want any hints on the exercise. But I don't know why, I am just totally confused on this:
Is it enough to show that if I have a structure of a k-linear field the I get the module category AND if I have a module category I get the structure of a k-linear category? Or do I then have to show that if I compose those two construction I get the same as before? does that even make sense?
I am so confused right now, I feel like I should know this but maybe my brain just gave up :P
One interpretation of "is the same as" is to show there exists a bijection between the sets of such structures. This corresponds to your proposal in the post, in the version including "composition": from one you get a unique choice of he other. A more refined interpretation would be that there exists an equivalence of categories, or even of 2-categories, between the collection of all semisimple $k$-linear categories and the collection of all semisimple $k-\mathrm{vector}$-module categories.