Enriched functor

In category theory, an enriched functor is a variant on a special type of mapping between categories.

Definition

A functor T is said to be C-enriched if for all objects X and Y in C, there are arrows

$tXY:(X \rightarrow Y) \longrightarrow (TX \rightarrow TY)$

satisfying

$tXX(\mathrm{id}(X)) = \mathrm{id}(TX)\,$

for all X in C, and

$tXY(f) \circ tYZ(g) = tXZ(f \circ g)$

for all f: X → Y and g: Y → Z in C.