Theorem: Right Adjoint Functor is Unique up to Natural Isomorphism

Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let  $\tuple {F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Then:
:the right adjoint functor $G$ of $F$ is unique up to isomorphism in the functor category $\operatorname{Funct} \tuple{C, D}$.
That is:
:right adjoint functor $G$ is unique up to natural isomorphism.

Proof:
Let $\tuple {F, G, \alpha}$ and $\tuple {F, H, \gamma}$ be adjunctions between $\mathbf C$ and $\mathbf D$.
From Characterization of Adjunction Using Counit of Adjunction:
:there exists a natural transformation $\xi: FG \to \operatorname {id}_{\mathbf C}$ such that:
::for each object $C$ in $\mathbf C$ the morphism $\xi_C:FGC \to C$ is a universal morphism from functor $F$ to object $GC$
and:
:there exists a natural transformation $\phi: FG \to \operatorname {id}_{\mathbf C}$ such that:
::for each object $C$ in $\mathbf C$ the morphism $\phi_C:FHC \to C$ is a universal morphism from functor $F$ to object $HC$
From Universal Morphism from Functor to Object is Unique up to Isomorphism:
:for each object $C$ in $\mathbf C$ there exists an isomorphism $\theta_C:HC \to GC \in \mathbf D$:
::$(1):\quad \phi_C = \xi_C \circ F \theta_C$
It will be shown that $\theta: H \to G$ is a natural isomorphism.
Let $g: C_1 \to C_2 \in \mathbf C$ be a morphism.
By definition of natrual transformation:
:$(2):\quad \xi_{C_2} \circ FGg = g \circ \xi_{C_1}$
:$(3):\quad \phi_{C_2} \circ FHg = g \circ \phi_{C_1}$
From $(1)$, $(2)$ and $(3)$ the following diagram commutes:
::<nowiki>$\begin{xy}
<0em,10em>*+{FGC_1} = "TL",
<18em,10em>*+{FGC_2} = "TR",
<4.5em,5em>*+{C_1} = "ML",
<13.5em,5em>*+{C_2} = "MR",
<0em,0em>*+{FHC_1} = "BL",
<18em,0em>*+{FHC_2} = "BR",
<1.5em,5em>*+{(1)} = "L",
<9em,2.5em>*+{(3)} = "T",
<9em,8em>*+{(2)} = "B",
<16.5em,5em>*+{(1)} = "R",
"TL";"TR" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{FGg},
"BL";"BR" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{FHg},
"ML";"MR" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g},
"BL";"TL" **@{-} ?>*@{>} ?<>(.5)*!/_2em/{F\theta_{C_1} },
"BR";"TR" **@{-} ?>*@{>} ?<>(.5)*!/^2em/{F\theta_{C_2} },
"TL";"ML" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{\xi_{C_1} },
"BL";"ML" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{\phi_{C_1} },
"TR";"MR" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{\xi_{C_2} },
"BR";"MR" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{\phi_{C_2} },
\end{xy}$</nowiki>
We have:
{{begin-eqn}}
{{eqn | l = g \circ \phi_{C_1}
      | r = g \circ \xi_{C_1} \circ F \theta_{C_1}
      | c = By $(1)$
}}
{{eqn | r = \xi_{C_2} \circ FGg \circ F \theta_{C_1}
      | c = By $(2)$
}}
{{eqn | r = \xi_{C_2} \circ \map F {Gg \circ \theta_{C_1} }
      | c = {{Defof|Covariant Functor}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = g \circ \phi_{C_1}
      | r = \phi_{C_2} \circ FHg
      | c = By $(3)$
}}
{{eqn | r = \xi_{C_2} \circ F \theta_{C_2} \circ FHg
      | c = By $(1)$
}}
{{eqn | r = \xi_{C_2} \circ \map F {\theta_{C_2} \circ Hg}
      | c = {{Defof|Covariant Functor}}
}}
{{end-eqn}}
By universality of $\xi_{C_2}$ applied to $g \circ \phi_{C_1}$:
:$\exists ! f: HC_1 \to GC_2 \in \mathbf D : \xi_{C_2} \circ \map F f = g \circ \phi_{C_1}$
Hence:
:$\theta_{C_2} \circ Hg = Gg \circ \theta_{C_1}$
It follows that $\theta$ is a natural isomorphism by definition.
{{qed}}
Category:Adjunctions
Category:Natural Isomorphisms
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