Existence of ind-right adjoint functor for semi-simple category?

Question

I'm just reading a lemma in Yves ANDRE's seminar on finite dimensional motives.

Soit $Σ:Rep_F G→T$ un ⊗-foncteur vers une categorie F-tensorielle T ……

where $G$ is a pro-reductive group scheme, $Rep_F G$ is the category of finite dimensional representations of $G$, and an "F-tensorielle" category means a pseudo-abelian F-linear rigid ⊗-category.

In the proof of this lemma, the author says since $Rep_F G$ is semi-simple, there exists a functor right ind-adjoint to $\Sigma$, that is a functor $\Theta:T→REP_F G$ from $T$ to the category of representations (not necessary finite dimensional) of $G$ which satisfies the condition analogue to that of adjoint functors.

Why can we find such functor, and how can we construct if?

Answer 1

The functor $\Sigma$ admits an ind-adjoint if and only if it is left exact (SGA 4, Exp. I, 8.11.4). Every short exact sequence in a semisimple abelian category splits, and $\Sigma$ commutes with finite direct sums (since it is by assumption $F$-linear), so the conclusion follows.