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 $C(x, y)$ denotes the hom-object in $V$,
such that the following diagrams commute for all $a, b, c \in C_0$:
respect for composition:
respect for units:
$V$‘s unit object $I$ and the family of identity elements $J_i$ from it are defined in enriched category.
Emily Riehl, chapter 3 Basics of enriched category theory in Categorical Homotopy Theory
For more references see at enriched category theory.
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