In the usual theory of categories, with any two objects A, B of a category A there is associated a set A (A B) of morphisms of A into B. Frequently the set A (A B) is endowed with an additional structure such as a privileged element or an abelian group structure. It has become clear that as the ramifications of the theory of categories increase, the structures that A (A B) will carry will be richer and more complex. The need for a general theory has been widely felt for some time, and beginnings have been made in various directions and often under restrictive hypotheses; e.g. by Mac Lane , Kelly , Bénabou , Linton 
KeywordsNatural Transformation Natural Isomorphism Monoidal Category Follow Diagram Commute Monoidal Structure
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