Theorem: Characterization of Adjunction via Triangle Identities/Sufficient 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 $\eta: \operatorname {id}_{\mathbf D} \to GF$ and $\xi: FG \to \operatorname {id}_{\mathbf C}$ be natural transformations such that:
:$G \xi \circ \eta G = \operatorname{id}_G$
:$\xi F \circ F \eta = \operatorname{id}_F$
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>
Then:
:there exists an adjunction $\tuple{F, G, \alpha}$ between $\mathbf C$ and $\mathbf D$:
::$\eta$ is the unit of adjunction $\alpha$
::$\xi$ is the counit of adjunction $\alpha$.

Proof:
For objects $D \in \mathbf D$ and $C \in \mathbf C$ let:
:$\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$
:$\beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, C}$
be the mappings defined by:
:$\forall g \in \map {\mathrm {Hom}_{\mathbf C} } {FD, C}: \map {\alpha_{\tuple{D, C}}} g = Gg \circ \eta_D$
:$\forall f \in \map {\mathrm {Hom}_{\mathbf D} } {D, GC}: \map {\beta_{\tuple{D, C}}} f = \xi_C \circ Ff$
From Characterization of Adjunction Using Right Adjuncts of Morphisms:
::the triple $\tuple {F, G, \alpha}$ is an adjunction
{{iff}}:
:$(1)\quad$ for objects $D \in \mathbf D$ and $C \in \mathbf C$:
::$\qquad\qquad\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$ is a bijection
:$(2)\quad$ for every $g:FD \to C \in \mathbf C$:
::$\qquad\qquad\map {\alpha_{\tuple{D, C} } } {g} = Gg \circ \map {\alpha_{\tuple{D, FD} } } {\operatorname{id}_{FD}}$
:$(3)\quad$ for objects $D_1, D_2 \in \mathbf D$ and morphism $f:D_1 \to D_2 \in \mathbf D$:
::$\qquad\qquad\map {\alpha_{\tuple{D_1, FD_2} } } {Ff} = \map {\alpha_{\tuple{D_2, FD_2} } } {\operatorname{id}_{FD_2}} \circ f$
==== Lemma 1 ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 1}}{{qed|lemma}}
==== Lemma 2 ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 2}}{{qed|lemma}}
==== Lemma 3 ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 3}}{{qed|lemma}}
==== Lemma 4 ====
{{:Characterization of Adjunction via Triangle Identities/Lemma 4}}{{qed|lemma}}
From Characterization of Adjunction Using Right Adjuncts of Morphisms:
:$\tuple{F, G, \alpha}$ is an adjunction between $\mathbf C$ and $\mathbf D$.
From Lemma 1:
:$\eta$ is the unit of adjunction $\alpha$ by definition.
:$\xi$ is the counit of adjunction $\alpha$ by definition.