Theorem: Characterization of Adjunction via Triangle Identities/Necessary Condition

Let $\mathbf C$, $\mathbf D$ be locally small categories.
Let  $F: \mathbf D \to \mathbf C$ and $G:\mathbf C \to \mathbf D$ be covariant functors.
Let $\tuple{F, G, \alpha}$ be an adjunction between $\mathbf C$ and $\mathbf D$.
Let $\eta$ denote the unit of adjunction $\alpha$
Let $\xi$ denote the counit of adjunction $\alpha$.
Then:
:$G \xi \circ \eta G = \operatorname{id}_G$
:$\xi F \circ F \eta = \operatorname{id}_F$
where:
:$G \xi, F \eta$ are right whiskered composites
:$\eta G, \xi F$ are left whiskered composites
:$\operatorname{id}_G, \operatorname{id}_F$ are identity natural transformations
That is, the following diagrams commute in the functor categories $\map {\operatorname{Funct} } {\mathbf C, \mathbf D}$ and $\map {\operatorname{Funct} } {\mathbf D, \mathbf C}$ respectively:
<nowiki>$\begin{xy}
<7em,5em>*+{GFG} = "TC",
<2.5em,0em>*+{G} = "BL",
<11.5em,0em>*+{G} = "BR",
<7em, -4em>*+{\map {\operatorname{Funct} } {\mathbf C, \mathbf D}} = "BBC",
"BL";"BR" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{\operatorname{id}_G},
"BL";"TC" **@{-} ?>*@{>} ?<>(.4)*!/_1em/{\eta G},
"TC";"BR" **@{-} ?>*@{>} ?<>(.6)*!/_1em/{G \xi},
<23.5em,5em>*+{FGF} = "TC",
<19em,0em>*+{F} = "BL",
<28em,0em>*+{F} = "BR",
<23.5em, -4em>*+{\map {\operatorname{Funct} } {\mathbf D, \mathbf C}} = "BBC",
"BL";"BR" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{\operatorname{id}_F},
"BL";"TC" **@{-} ?>*@{>} ?<>(.4)*!/_1em/{F \eta},
"TC";"BR" **@{-} ?>*@{>} ?<>(.6)*!/_1em/{\xi F},
\end{xy}$</nowiki>

Proof:
By definition of unit and counit of adjunction:
:$\eta$ and $\xi$ are natural transformations.
==== Proof of $G \xi \circ \eta G = \operatorname{id}_G$ ====
We have:
{{begin-eqn}}
{{eqn | q = \forall C \in \mathbf C
      | l = \paren{G \xi \circ \eta G}_C
      | r = \paren{G \xi}_C \circ \paren{\eta G}_C
      | c = {{Defof|Vertical Composition of Natural Transformations|Vertical Composition}}
}}
{{eqn | r = G \xi_C \circ \paren{\eta G}_C
      | c = {{Defof|Right Whiskered Composite of Natural Transformation|Right Whiskered Composition}}
}}
{{eqn | r = G \xi_C \circ \eta_{GC}
      | c = {{Defof|Left Whiskered Composite of Natural Transformation|Left Whiskered Composition}}
}}
{{eqn | r = G \xi_C \circ \map {\alpha_{\tuple{GC, FGC} } } {\operatorname {id}_{FGC} }
      | c = {{Defof|Unit of Adjunction}}
}}
{{eqn | r = \map {\alpha_{\tuple{GC, C} } } {\xi_C \circ \operatorname {id}_{FGC} }
      | c = Characterization of Adjunction Using Right Adjuncts of Compositions
}}
{{eqn | r = \map {\alpha_{\tuple{GC, C} } } {\xi_C}
      | c = {{Defof|Identity Morphism}}
}}
{{eqn | r = \map {\alpha_{\tuple{GC, C} } } {\map {\beta_{\tuple{GC, C} } } {\operatorname {id}_{GC} } }
      | c = {{Defof|Counit of Adjunction}}
}}
{{eqn | r = \operatorname {id}_{GC}
      | c = {{Defof|Inverse Mapping}}
}}
{{eqn | r = \paren{\operatorname {id}_G}_C
      | c = {{Defof|Identity Natural Transformation|Identity Natural Transformation of $G$}}
}}
{{end-eqn}}
It follows that:
:$G \xi \circ \eta G = \operatorname{id}_G$
{{qed|lemma}}
For each object $\tuple{D, C} \in \mathbf D \times \mathbf C$ let $\beta_{\tuple{D, C} }$ denote the inverse of $\alpha_{\tuple{D, C} }$:
:$\beta_{\tuple{D, C} } = \alpha_{\tuple{D, C} }^{-1} : \map {\operatorname{Hom}_{\mathbf D} } {D, GC} \to \map {\operatorname{Hom}_{\mathbf C} } {FD, C}$
==== Proof of $\xi F \circ F \eta = \operatorname{id}_F$ ====
We have:
{{begin-eqn}}
{{eqn | q = \forall D \in \mathbf D
      | l = \paren{\xi F \circ F \eta}_D
      | r = \paren{\xi F}_D \circ \paren{F \eta}_D
      | c = {{Defof|Vertical Composition of Natural Transformations|Vertical Composition}}
}}
{{eqn | r = \paren{\xi F}_D \circ F \eta_D
      | c = {{Defof|Right Whiskered Composite of Natural Transformation|Right Whiskered Composition}}
}}
{{eqn | r = \xi_{FD} \circ F \eta_D
      | c = {{Defof|Left Whiskered Composite of Natural Transformation|Left Whiskered Composition}}
}}
{{eqn | r = \map {\beta_{\tuple{GFD, FD} } } {\operatorname {id}_{FD} } \circ F \eta_D
      | c = {{Defof|Counit of Adjunction}}
}}
{{eqn | r = \map {\beta_{\tuple{D, FD} } } {\operatorname {id}_{FD} \circ \eta_D } 
      | c = Characterization of Adjunction Using Left Adjuncts of Compositions
}}
{{eqn | r = \map {\beta_{\tuple{D, FD} } } {\eta_D }
      | c = {{Defof|Identity Morphism}}
}}
{{eqn | r = \map {\beta_{\tuple{D, FD} } } {\map {\alpha_{\tuple{D, F D} } } {\operatorname {id}_{FD} } }
      | c = {{Defof|Unit of Adjunction}}
}}
{{eqn | r = \operatorname {id}_{FD}
      | c = {{Defof|Inverse Mapping}}
}}
{{eqn | r = \paren{\operatorname {id}_F}_D
      | c = {{Defof|Identity Natural Transformation|Identity Natural Transformation of $G$}}
}}
{{end-eqn}}
It follows that:
:$\xi F \circ F \eta = \operatorname{id}_F$
{{qed|lemma}}
The result follows.