 Mate correspondence

ID: mate-correspondence

Codex  0 Created 2026-10-03 Updated 2026-10-14

For adjunctions

F_{i} ⊣ G_{i}

, taking mates gives a natural bijection

Nat (F_{1}, F_{2}) ≅ Nat (G_{2}, G_{1}) .

α : F_{1} \to F_{2}

, its right mate is

G_{2} η_{1} G_{2} G_{1} F_{1} G_{2} G_{1} α G_{2} G_{1} F_{2} G_{2} G_{1} ε_{2} G_{1} .

Taking mates reverses vertical composition.

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