Generalising connected components

Let I:C→M be a reflection of a category C with pullbacks into a full subcategory M of C. We introduce an additional structure on C involving a pullback-preserving functor U:C→S, which allows us to prove that the reflection I 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 S is the category of sets.