Let $G$ and $G'$ be [quivers](https://ncatlab.org/nlab/show/quiver). If their [path categories](https://ncatlab.org/nlab/show/path+category#free_category_on_a_directed_graph) $Path[G]$ and $Path[G']$ are isomorphic, does is follow that $G$ is isomorphic to $G'$?



Does the path category of a quiver determine the quiver up to isomorphism?
Yes. Given a Quiver $G$, you can identify $G$ as the subquiver of $Path[G]$ of arrows that are not identity and cannot be written as composite of non-identity arrows. So any isomorphism between $Path[G]$ and $Path[G']$ send elements of $G$ to elements of $G'$ and restrict to an isomorphism between $G$ and $G'$.


