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.