Item 9 (00W5) — The Clowder Project

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Descending Functions to Quotient Sets, VI. Let $R$ be a relation on $X$ and let $\mathord {\sim }^{\mathrm{eq}}_{R}$ be the equivalence relation associated to $R$. The following conditions are equivalent:
1. The map $f$ satisfies the equivalent conditions of Item 4:
  - There exists a map
    \[ \overline{f}\colon X/\mathord {\sim }^{\mathrm{eq}}_{R}\to Y \]
    
    making the diagram
    
    commute.
  - For each $x,y\in X$, if $x\sim ^{\mathrm{eq}}_{R}y$, then $f\webleft (x\webright )=f\webleft (y\webright )$.
2. For each $x,y\in X$, if $x\sim _{R}y$, then $f\webleft (x\webright )=f\webleft (y\webright )$.

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