In [the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik](http://www-math.mit.edu/~etingof/egnobookfinal.pdf) 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.



Any exact faithful functor is represented by a unique projective generator
Let $\mathcal C$ be the category of finite dimensional left modules over a finite dimensional ring $R$. Let $G: \mathcal C \to \mathrm{Vec}$ be an exact and faithful functor to finite dimensional vector spaces. We use $V^\*$ to denote the dual vector space. For motivation, notice that if we had a representing object $M$, we would have $$G(R^\*) = Hom\_R(M,R^\*) = Hom\_R(R, M^\*) = M^\*.$$


Now $G(R^\*)$ is a right $R$ module via the action of $R$ by left multiplication. So we define $P:= G(R^\*)^\*$ to be the dual left module and consider the functor $Hom\_R(P,-)$. 


This functor is tautologically left exact, and it takes the injective left module $R^\*$ to $Hom\_R(P,R^\*) = Hom\_R(R, G(R^\*)) = G(R^\*)$. Any other finite left module $M$ admits an injective presentation $$0 \to M \to R^{\* \oplus a} \to R^{\* \oplus b} \to $$
dual to the presentation of $M^\*$ as a right $R$ module. So by left exactness, we see that there is a natural isomorphism $Hom\_R(P,-) \simeq G(-)$. 


Thus $G$ is represented by $P$. Since $G$ is right exact $P$ is projective, and since it is faithful $P$ is a generator.


