Lemma 97.13.1. The functor in groupoids $FA_ d$ defined in (97.13.0.2) is isomorphic (!) to the functor in groupoids which associates to a scheme $T$ the category with

set of objects is $X(T)$,

set of morphisms is $G(T) \times X(T)$,

$s : G(T) \times X(T) \to X(T)$ is the projection map,

$t : G(T) \times X(T) \to X(T)$ is $a(T)$, and

composition $G(T) \times X(T) \times _{s, X(T), t} G(T) \times X(T) \to G(T) \times X(T)$ is given by $((g, m), (g', m')) \mapsto (gg', m')$.

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