In a concrete category, an injective hull of an object is an extension of such that is injective and is an essential embedding?. It is the dual concept to projective cover.
In general, there is no way of making the assignment of the injective hull to an object into a functor such that there is a natural transformation between the identity functor and that functor.
Given a class of objects in a category, an -hull (or -envelope) of an object is a map such that the following two conditions hold:
Any map to an object in factors through via some map .
Whenever a map satisfies then it must be an automorphism.
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution