ScalingStacks

Proposition 7.4.5. Let $Z$ be a curve with a finite admissible relation $\sim$ and let $q:Z\to Z/\!\!\sim$ be the quotient map. The functor $q^{\#}:\operatorname{add}\nolimits({\mathcal{P}}(Z/\!\!\sim))\to\operatorname{add}\nolimits({\mathcal{P}}_{q}(Z))$ is faithful and every map in ${\mathcal{P}}^{\bullet}(Z/\!\!\sim)$ is in the image of the functor $q:{\mathcal{P}}_{q}^{\bullet}(Z)\to{\mathcal{P}}^{\bullet}(Z/\!\!\sim)$.