Pointed classification of connected groupoid coverings

Send $Ha$ to the endpoint of the lift of $a$ from $e$. The identity lifts to $1_e$, so $H$ is sent to $e$. The endpoint criterion says precisely that this is well defined and injective on the objects over each target; all targets are retained, so it is injective on all objects. Connectedness of $E$ gives an arrow from $e$ to every object, so this map is surjective. Send $(Ha,g)$ to the unique lift of $g$ from the assigned object. Product lifting shows its target is the assigned object of $Hag$, and that this defines a functor over $G$. It is bijective on arrows because it is bijective on objects and on each source star. Its inverse therefore also preserves products and identities, making it an isomorphism.

Any pointed functor over $G$ must send every lift of $a$ from $H$ to the lift of $a$ from $e$, so it has the same objects and arrows. This proves uniqueness. An isomorphism preserving basepoints preserves the image of the vertex group, proving necessity and sufficiency of equality. If a basepoint changes along the lift of $a\in G_x$, conjugation gives the new image $a^{-1}Ha$: conjugate loops by the lifted arrow and apply the functor, in both directions. Every point above $x$ is reached by such a lift by connectedness. Hence an unpointed isomorphism gives conjugacy; conversely conjugacy permits choosing a point with the required image subgroup, after which pointed classification supplies the isomorphism.