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

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

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

The theorem is added to my book.