Item 4 (00W0) — The Clowder Project

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Descending Functions to Quotient Sets, I. Let $R$ be an equivalence relation on $X$. The following conditions are equivalent:
1. There exists a map
  \[ \overline{f}\colon X/\mathord {\sim }_{R}\to Y \]
  
  making the diagram
  
  commute.
2. We have $R\subset \mathrm{Ker}\webleft (f\webright )$.
3. For each $x,y\in X$, if $x\sim _{R}y$, then $f\webleft (x\webright )=f\webleft (y\webright )$.

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