Theorem: Characterization of Adjunction via Triangle Identities/Lemma 2

:for objects $D \in \mathbf D$ and $C \in \mathbf C$:
::$\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$ is a bijection 
:with inverse:
::$\beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, C}$

Proof:
Let $D \in \mathbf D$ and $C \in \mathbf C$ be objects.
Let $g: FD \to C \in \mathbf C$ and $f: D \to GC \in \mathbf D$ be morphisms
By definition of natural transformation:
:$(1): \quad GF f \circ \eta_D = \eta_{GC} \circ f$
and:
:$(2): \quad g \circ \xi_{FD} = \xi_C \circ FG g$
We have:
{{begin-eqn}}
{{eqn | l = \map {\paren{\alpha_{\tuple{D, C} } \circ \beta_{\tuple{D, C} } } } f
      | r = \map {\alpha_{\tuple{D, C} } } {\map {\beta_{\tuple{D, C} } } f}
      | c = {{Defof|Composite Mapping}}
}}
{{eqn | r = \map {\alpha_{\tuple{D, C} } } {\xi_C \circ Ff}
      | c = Definition of $\beta_{\tuple{D, C} }$
}}
{{eqn | r = \map G {\xi_C \circ Ff} \circ \eta_D
      | c = Definition of $\alpha_{\tuple{D, C} }$
}}
{{eqn | r = G \xi_C \circ GF f \circ \eta_D
      | c = {{Defof|Covariant Functor}}
}}
{{eqn | r = G \xi_C \circ \eta_{GC} \circ f
      | c = By $(1)$
}}
{{eqn | r = \paren{G \xi}_C \circ \eta_{GC} \circ f
      | c = {{Defof|Right Whiskered Composite of Natural Transformation|Right Whiskered Composite}}
}}
{{eqn | r = \paren{G \xi}_C \circ \paren{\eta G}_C \circ f
      | c = {{Defof|Left Whiskered Composite of Natural Transformation|Left Whiskered Composite}}
}}
{{eqn | r = \paren{G \xi \circ \eta G}_C \circ f
      | c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{eqn | r = \paren{\operatorname{id}_G}_C \circ f
      | c = {{hypothesis}}
}}
{{eqn | r = \operatorname{id}_{GC} \circ f
      | c = {{Defof|Identity Natural Transformation}}
}}
{{eqn | r = f
      | c = {{Defof|Identity Morphism}}
}}
{{end-eqn}}
and:
{{begin-eqn}}
{{eqn | l = \map {\paren{\beta_{\tuple{D, C} } \circ \alpha_{\tuple{D, C} } } } f
      | r = \map {\beta_{\tuple{D, C} } } {\map {\alpha_{\tuple{D, C} } } f}
      | c = {{Defof|Composite Mapping}}
}}
{{eqn | r = \map {\beta_{\tuple{D, C} } } {Gg \circ \eta_D}
      | c = Definition of $\alpha_{\tuple{D, C} }$
}}
{{eqn | r = \xi_C \circ \map F {Gg \circ \eta_D}
      | c = Definition of $\beta_{\tuple{D, C} }$
}}
{{eqn | r = \xi_C \circ FG g \circ F \eta_D
      | c = {{Defof|Covariant Functor}}
}}
{{eqn | r = g \circ \xi_{FD} \circ F \eta_D
      | c = By $(2)$
}}
{{eqn | r = g \circ \xi_{FD} \circ \paren{F \eta}_D
      | c = {{Defof|Right Whiskered Composite of Natural Transformation|Right Whiskered Composite}}
}}
{{eqn | r = g \circ \paren{\xi F}_D \circ \paren{F \eta}_D
      | c = {{Defof|Left Whiskered Composite of Natural Transformation|Left Whiskered Composite}}
}}
{{eqn | r = g \circ \paren{\xi F \circ F \eta}_D
      | c = {{Defof|Vertical Composition of Natural Transformations}}
}}
{{eqn | r = g \circ \paren{\operatorname{id}_F}_D
      | c = {{hypothesis}}
}}
{{eqn | r = g \circ \operatorname{id}_{FD}
      | c = {{Defof|Identity Natural Transformation}}
}}
{{eqn | r = g
      | c = {{Defof|Identity Morphism}}
}}
{{end-eqn}}
It follows that for objects $D \in \mathbf D$ and $C \in \mathbf C$:
:$\alpha_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf C} } {FD, C} \to \map {\mathrm {Hom}_{\mathbf D} } {D, GC}$ is a bijection 
with inverse:
:$\beta_{\tuple{D, C}}: \map {\mathrm {Hom}_{\mathbf D} } {D, GC} \to \map {\mathrm {Hom}_{\mathbf C} } {FD, C}$
{{qed}}
Category:Characterization of Adjunction via Triangle Identities
oh371ffevttm454jm14uremv6mly4r4