Let $C$ be a $k$-linear abelian category in the sense of Deligne & Milne's notes, in which $k$ is a field. Proposition 1.20 in the book states that give such a category together with an exact faithful functor $F: C \to \mathrm{Vect}_k$ (the category of finite dimensional vector spaces over $k$), then $C$ itself is a rigid abelian tensor category.
The authors did not write a full proof but just assert that it can be proved directly without referring to the next section in which they prove $C$ is isomorphic to the category $\mathrm{Rep}_k(G)$ of finite dimensional representations of a $k$-group scheme $G$. In addition, the note that the condition:
if $F(L)$ has dimension $1$, then there exists an object $L \otimes L^{-1} = U$.
is sufficient to show that $C$ is rigid. I don't even understand why should we need this condition. I tried to demonstrate the existence of the dual (hence, the internal hom) $X^{\vee}$ for every object $X$ but it seems that the condition of $F$ being only faithful is not strong enough. Any suggestion on this would be appreciated!