Enriched category

Enriched functor

Let 𝖢,𝖣 be [[Enriched category|𝖬-categories]]. An 𝖬-functor¹ 𝐹 :𝖢 →𝖣 is the appropriate generalization of an ordinary functor. Thus it consists of

• A (class) function Ob⁡(𝐹) :Ob⁡(𝖢) →Ob⁡(𝖣) :𝑎 ↦𝐹𝑎;
• For every ordered pair 𝑎,𝑏 ∈Ob⁡(𝖢) a morphism 𝐹𝑎,𝑏 :𝖢(𝑎,𝑏) →𝖣(𝐹𝑎,𝐹𝑏)

such that these data respect composition

and unitality

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Footnotes

1. Also called a functor enriched over 𝖬. ↩