After a little more thought, I have however proved (see addons.pdf) a cute algebraic description of normal quasi-uniformities:

Theorem The following are pairwise equivalent:

Endoreloid is normal.

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I continue to investigate the article by Taras Banakh, Alex Ravsky, how can it be generalized with reloids instead of uniform spaces and funcoids instead of topological spaces.

The proof was with an error, see http://www.mathematics21.org/binaries/addons.pdf