A new (but easy to prove) theorem in my research book:

Theorem Let \mu and \nu be endomorphisms of some partially ordered dagger precategory and f\in\mathrm{Hom}(\mathrm{Ob}\mu;\mathrm{Ob}\nu) be a monovalued, entirely defined morphism. Then f\in\mathrm{C}(\mu;\nu)\Leftrightarrow f\in\mathrm{C}(\mu^{\dagger};\nu^{\dagger}).

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