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



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> Soit $Σ:Rep\_F G→T$ un ⊗-foncteur vers une categorie F-tensorielle T ……
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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?



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


