• 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 $f\webleft (x\webright )=f\webleft (y\webright )$.


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