ScalingStacks

Proposition 7.4.3. The functor $f:{\mathcal{S}}^{\bullet}(Z,1)\to{\mathcal{S}}^{\bullet}(Z^{\prime},1)$ defines a faithful pointed functor

$$f:{\mathcal{P}}^{\bullet}_{f}(Z)\to{\mathcal{P}}^{\bullet}(Z^{\prime}),\ I\mapsto f(I),\ \theta\mapsto(f(\theta_{s}))_{f(s)}.$$

In particular if $f:Z\to Z^{\prime}$ is injective then we have a faithful pointed functor $f:{\mathcal{P}}^{\bullet}(Z)\to{\mathcal{P}}^{\bullet}(Z^{\prime})$.