Definition 89.19.5. Let $p : \mathcal{F} \to \mathcal{C}$ be a category cofibered in groupoids over an arbitrary base category $\mathcal{C}$. Assume a choice of pushforwards has been made. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F})$ and let $U = p(x)$. Let $U/\mathcal{C}$ denote the category of objects under $U$. The automorphism functor of $x$ is the functor $\mathit{Aut}(x) : U/\mathcal{C} \to \textit{Sets}$ sending an object $f : U \to V$ to $\text{Aut}_ V(f_*x)$ and sending a morphism

$\xymatrix{ V' \ar[rr] & & V\\ & U \ar[ul]^{f'} \ar[ur]_ f & }$

to the homomorphism $\text{Aut}_{V'}(f'_*x) \to \text{Aut}_ V(f_*x)$ coming from the unique morphism $f'_*x \to f_*x$ lying over $V' \to V$ and compatible with $x \to f'_*x$ and $x \to f_*x$.

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