A new theorem about generalized continuity

I had this theorem in mind for a long time, but formulated it exactly and proved only yesterday.

Theorem f \in \mathrm{C} (\mu \circ \mu^{- 1} ; \nu \circ \nu^{- 1}) \Leftrightarrow f \in \mathrm{C} (\mu; \nu) for complete endofuncoids \mu, \nu and principal monovalued and entirely defined funcoid f \in \mathsf{FCD} (\mathrm{Ob}\, \mu; \mathrm{Ob}\, \nu), provided that \mu is reflexive, and \nu is T_1-separable.

Here f \in \mathrm{C} (\mu; \nu) means that f is a continuous morphism from a “space” \mu to a “space” \nu.

The theorem is added to my book.

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