Enriched Functors
An enriched functor is the appropriate generalization of the notion of a functor to enriched categories. Enriched functors are then maps between enriched categories which respect the enriched structure.
If C and D are M-categories (that is, categories enriched over monoidal category M), an M-enriched functor T: C → D is a map which assigns to each object of C an object of D and for each pair of objects a and b in C provides a morphism in M Tab: C(a,b) → D(T(a),T(b)) between the hom-objects of C and D (which are objects in M), satisfying enriched versions of the axioms of a functor, viz preservation of identity and composition.
Because the hom-objects need not be sets in an enriched category, one cannot speak of a particular morphism. There is no longer any notion of an identity morphism, nor of a particular composition of two morphisms. Instead, morphisms from the unit to a hom-object should be thought of as selecting an identity and morphisms from the monoidal product should be thought of as composition. The usual functorial axioms are replaced with corresponding commutative diagrams involving these morphisms.
In detail, one has that the diagram
commutes, which amounts to the equation
where I is the unit object of M. This is analogous to the rule F(ida) = idF(a) for ordinary functors. Additionally, one demands that the diagram
commute, which is analogous to the rule F(fg)=F(f)F(g) for ordinary functors.
Read more about this topic: Enriched Category
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