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Given a quasi-compact category fibered in groupoids $\mathcal{X}$ and amonoidal subcategory $\mathcal{C}$ of its category of locally free sheaves$\text{Vect}(\mathcal{X})$, we are going to introduce the stack of fiberfunctors $\text{Fib}_{\mathcal{X},\mathcal{C}}$ with source $\mathcal{C}$,which comes equipped with a map$\mathcal{P}_{\mathcal{C}}\colon\mathcal{X}\to\text{Fib}_{\mathcal{X},\mathcal{C}}$and a functor$\mathcal{G}\colon\mathcal{C}\to\text{Vect}(\text{Fib}_{\mathcal{X},\mathcal{C}})$.  If $\mathcal{C}$ generates $\text{QCoh}(\mathcal{X})$ and $\mathcal{X}$ is anfpqc stack with quasi-affine diagonal, we show that$\mathcal{P}_{\mathcal{C}}\colon\mathcal{X}\to\text{Fib}_{\mathcal{X},\mathcal{C}}$is an equivalence, as it happens by Tannaka's reconstruction when $\mathcal{X}$is an affine gerbe over a field. In general, under mild assumption on$\mathcal{C}$, e.g. $\mathcal{C}=\text{Vect}(\mathcal{X})$, we show that$\text{Fib}_{\mathcal{X},\mathcal{C}}$ is a quasi-compact fpqc stack withaffine diagonal and that the image $\mathcal{G}(\mathcal{C})$ generates$\text{QCoh}(\text{Fib}_{\mathcal{X},\mathcal{C}})$.

Stacks of fiber functors and Tannaka's reconstruction