In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says:

'Conversely, it is well known (and easy to show) that any exact faithful functor $F : \mathcal{C} \rightarrow \text{Vec}$ is represented by a unique (up to a unique isomorphism) projective generator $P$.'

But I could not find any proof of that fact. Can someone tell me how to prove it or where I can find a proof?

Note: Here $\mathcal{C}$ is a finite k-linear abelian category for some field k.