Full faithfulness of the global-space functor for finite orbispaces

Let $\mathrm{OrbSpc}^f$ be the category of finite orbispaces and let $\mathrm{GloSpc}$ be the category of global spaces. For a faithful vector bundle $E$ over $X$, define the associated global space by

$$X\longmapsto\operatorname{Emb}_X(E,-).$$

Global-space full-faithfulness conjecture. For $X\in\mathrm{OrbSpc}^f$ and $F\in\mathrm{GloSpc}$, the set $\operatorname{Hom}(X,F)$ coincides with the morphisms from the image of $X$ under this functor to $F$. Equivalently, the functor from finite orbispaces to global spaces is fully faithful. The conjecture would identify maps out of finite orbispaces with maps out of their associated global spaces.