Let be a reflection of a category with pullbacks into a full subcategory of . We introduce an additional structure on involving a pullback-preserving functor , which allows us to prove that the reflection is: (a) semi-left-exact if and only if it makes all connected components connected in an appropriate sense; (b) a reflection with stable units if and only if certain pullbacks of connected components are connected. This was previously done in the case where is the category of sets.