For a scheme $X$, denote by $\mathcal{Ell}_X[\text{isog}^{-1}]$ the category of elliptic curves on $X$ localized at isogenies. Consider the functor $$ \mathcal{Ell}^{isog}:Sch/S^{op}\rightarrow \text{Gpd}, \quad X \rightarrow \mathcal{Ell}_X[\text{isog}^{-1}]. $$ It was asked in this M.SE question if this was an algebraic stack. It is not, because

"forget isomorphism class and remember only isogeny class" map from the usual moduli space ought to be algebraic, but an algebraic map of curves has finite fibers but (over C, say) isogeny classes of non-isomorphic curves are infinite. (https://math.stackexchange.com/q/675207)

I'm wondering if the functor $\mathcal{Ell}^{isog}$ if not represented by an algebraic stack is a $2$-sheaf. I don't have any intuition about that, so I'd be glad for any hint.

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