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.
Examples
In Vect every object has an injective hull, . In other words, every vector space is already an injective object.