counit of an adjunction

余单位

Category Theory · Graduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: co-unit

Translation evidence: Literature usage

Meaning in context

For $F\dashv G$, the natural transformation $\varepsilon:FG\Rightarrow1_D$ whose component corresponds to $1_{GY}$ under the inverse adjunction bijection.

对于 F⊣G，由伴随双射的逆把 1_GY 对应到 ε_Y:FGY→Y 而组成的自然变换 ε:FG⇒1_D。

Example

For free vector spaces, $\varepsilon_V(\sum a_i e_{v_i})=\sum a_i v_i$.

Usage and distinctions

The counit and unit have different domains; triangle identities do not make their components literal inverses.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Emily Riehl — Category Theory in Context (current second-edition draft)§4.2, Lemma 4.2.3 · Checked 2026-10-14

Translation sources

1. 李文威 — 代数学方法，第一卷（作者源码）第2章，「伴随函子」，定义 def:adjunction-unit-counit「余单位」 · Checked 2026-10-14

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