ScalingStacks

Lemma 7.2.7. Let $f:Z\to Z^{\prime}$ be a morphism of curves.

Define an equivalence relation on $Z$ by $z_{1}\sim z_{2}$ if $f(z_{1})=f(z_{2})$. This is a finite relation on $Z$ and $f$ factors as a composition of morphisms of curves $Z\xrightarrow{f_{1}}Z/\!\!\sim\xrightarrow{f_{2}}Z^{\prime}$ where $f_{1}$ is the quotient map and $f_{2}$ is injective.