Could not include enriched category theory - contents
Enriched functors are used in place of functors in enriched category theory: like functors they send objects to objects, but instead of mapping hom-sets to hom-sets they assign morphisms in the enriching category between hom-objects, while being compatible with composition and units in the obvious way.
Given two categories $C, D$ enriched in a monoidal category $V$, an enriched functor $F: C \to D$ consists of
A function $F_0: C_0 \to D_0$ between the underlying collections of objects;
A $(C_0 \times C_0)$-indexed collection of morphisms of $V$,
[where denotes the hom-object in ], compatible with the enriched identities and compositions of $C$ and $D$;
such that the following diagrams commute for all $a, b, c \in C_0$:
respect for composition:
respect for units:
The standard reference on enriched category theory is