torsors from $P$ to $Q$ is simply a $G$-equivariant morphism of $X$-schemes or a generalized morphism covering the identity.

(2) If $R$ is a ring, $\rho:G\to H$ is an arrow of $({\mathbf{FGSch}}/R)$ and $P\to X$ is a $G$-torsor, we let $P{\times}^GH$ or $P{\times}^\rho H$ be the associated $H$-torsor (see [@demazure-gabriel70 III.4.3.2, p.368] or [@jantzen87 Part I, 5.14]).

(3) If $G$ is a flat group scheme over a ring $R$, $P\to X$ is a $G$-torsor and $M$ is a representation of $G$, then we let ${\theta}_P(M)$ stand for the coherent sheaf constructed by twisting $P$ by $M$, see [@jantzen87 Part 1, 5.8–9], where it is denoted by ${{\mathscr{L}}}_{P/G}(M)$.

Miscellaneous {#miscellaneous .unnumbered}
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(1) All tensor categories and functors are to be taken in the sense of [@deligne-milne82 §1]. Let $({\mathfrak{C}},{\otimes})$ be a rigid tensor category [@deligne-milne82 Definition 1.7]. If ${\mathfrak{C}}$ is additive, then, for $a=(a_1,\ldots,a_m)$ and $b=(b_1,\ldots,b_m)$ in ${\mathds{N}}^m$, and $E\in{\mathfrak{C}}$, we write ${\mathbf{T}}^{a,b}E$ to denote the object $\bigoplus_iE^{{\otimes}a_i}{\otimes}\check{E}^{{\otimes}b_i}$. If ${\mathfrak{C}}$ is in addition abelian, we let $\langle E\,;\,{\mathfrak{C}}\rangle_{\otimes}$ stand for the full subcategory of ${\mathfrak{C}}$ whose objects are subquotients of some ${\mathbf{T}}^{a,b}E$.

(2) If $X$ is a proper and reduced scheme over a field, we let ${\mathbf{EF}}(X)$ denote the category 