Enriched category

Enriched functor

Let $𝖢, 𝖣$ be $𝖬$ -categories. An $𝖬$ -functor¹ $𝐹 : 𝖢 \to 𝖣$ is the appropriate generalization of an ordinary functor. Thus it consists of

A (class) function $O b (𝐹) : O b (𝖢) \to O b (𝖣) : 𝑎 \mapsto 𝐹 𝑎$ ;
For every ordered pair $𝑎, 𝑏 \in O b (𝖢)$ a morphism $𝐹 𝑎, 𝑏 : 𝖢 (𝑎, 𝑏) \to 𝖣 (𝐹 𝑎, 𝐹 𝑏)$

such that these data respect composition

and unitality

#state/tidy | #lang/en | #SemBr

Also called a functor enriched over $𝖬$ . ↩