definition. preserve and reflect [kostecki2011introduction, 3.3] [tt-0018]

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A functor \(\mathscr {F}: {\cal C} \to {\cal D}\) is called to preserve a property \(\wp \) of an arrow iff for every \(f \in \operatorname {Arr}({\cal C})\) that has a property \(\wp \) it follows that \(\mathscr {F}(f) \in \operatorname {Arr}({\cal D})\) has this property. A functor \(\mathscr {F}: {\cal C} \to {\cal D}\) is called to reflect a property \(\wp \) of an arrow iff for every \(\mathscr {F}(f) \in \operatorname {Arr}({\cal D})\) that has a property \(\wp \) it follows that \(f \in \operatorname {Arr}({\cal C})\) has this property.