# When is a functor the associated bundle construction?

Let $$G$$ be a group and $$X$$ be a space. I am principally interested in these objects in the category of schemes over some base $$S$$, but an answer in another geometric category would be welcome.

Suppose I have a functor $$F$$ from $$Rep(G)$$ to bundles on $$X$$. What are necessary and sufficient conditions for $$F$$ to be the associated bundle construction for some principal $$G$$-bundle on $$X$$? Necessary conditions include that $$F$$ is an exact tensor functor and $$\mathrm{rank}\, F(V) = \dim V$$.

An answer is given in [1, §2.2] when $$S = spec(k)$$ and $$G/S$$ is an affine algebraic group. Let $$Rep(G)_{fd}$$ denote the category of finite-dimensional representations. $$F: Rep(G)_{fd}\to Qcoh(X)$$ must satisfy the following properties:

1. $$F$$ is $$k$$-additive and exact;
2. $$F$$ is a strict tensor functor;
3. $$F(\text{rank one trivial representation}) = \mathcal{O}_X$$;
4. $$F(V)$$ is locally free of rank $$\dim V$$ for all $$V$$.

One then extends $$F$$ to $$Rep(G)$$; then $$\mathbf{Spec}(F(\mathcal{O}_G))$$ is then a principal $$G$$-bundle whose associated bundle functor is $$F$$.

[1] Nori, Madhav V. "The fundamental group-scheme." Proc. Indian Acad. Sci. Math. Sci 91, no. 2 (1982): 73-122.