Adjoint functors
Uniqueness

If the functor F : D → C has two right adjoints G and G ′, then G and G ′ are naturally isomorphic. The same is true for left adjoints.

Conversely, if F is left adjoint to G, and G is naturally isomorphic to G ′ then F is also left adjoint to G ′. More generally, if 〈 F, G, ε, η〉 is an adjunction (with counit–unit (ε,η)) and

are natural isomorphisms then 〈 F ′, G ′, ε′, η′〉 is an adjunction where

Here ∘ {\displaystyle \circ } denotes vertical composition of natural transformations, and ∗ {\displaystyle \ast } denotes horizontal composition.
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