ScalingStacks

Lemma 7.1.13. Let $f:X\to X^{\prime}$ be a morphism of $1$-dimensional spaces. Define an equivalence relation on $X$ by $x_{1}\sim x_{2}$ if $f(x_{1})=f(x_{2})$. This defines a finite relation on $X$ and $f$ factors uniquely as a composition $f=\bar{f}\circ q$ where $\bar{f}:X/\!\!\sim\ \to X^{\prime}$ is a morphism of $1$-dimensional spaces and $q:X\to X/\!\!\sim$ is the quotient map.