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伴随函子的例子

Given functors {$F:D\rightarrow C$}, {$G:C\rightarrow D$}, and an adjunction {$F \dashv G$}, show in general and illustrate with examples that:

• {$C$} and {$D$} are categories.
  • Describe the objects.
  • Describe the morphisms.
  • Describe and verify the identity.
  • Verify the associativity of composition.
• {$F$} and {$G$} are functors.
  • Describe how they act on objects.
  • Describe how they act on morphisms.
  • Verify that they respect composition of morphisms.
• Show that {$\textrm{D}(F(C),D)\cong\textrm{C}(C,G(D))$} naturally in {$C\in\textrm{C}$}, {$D\in\textrm{D}$}. Demonstrate a natural isomorphism {$\Phi$} between the bifunctors {$\textrm{D}(F(\_),\_)$} and {$\textrm{C}(\_,G(\_))$}.

{$$\begin{matrix} & \textrm{D}(F(C),D) & \overset{\Phi_{C,D}}{\longrightarrow} & \textrm{C}(C,G(D)) \\ \textrm{D}(F(c),d) & \downarrow & & \downarrow & \textrm{D}(c,G(d)) \\ & \textrm{D}(F(C'),D')) & \overset{\Phi_{C',D'}}{\longrightarrow} & \textrm{C}(C',G(D')) \end{matrix}$$}

• Define the set function {$\Phi_{C,D}$}.
• Show that the set function {$\Phi_{C,D}$} is one-to-one.
• Show that the set function {$\Phi_{C,D}$} is onto.
• Define the set function {$\Phi^{-1}_{C,D}$}.
• Show that the set function {$\Phi^{-1}_{C,D}$} is one-to-one.
• Show that the set function {$\Phi^{-1}_{C,D}$} is onto.
• Show that {$\Phi^{-1}_{C,D}\Phi_{C,D}=\textrm{id}_{\textrm{D}(F(C),D)}$}.
• Show that {$\Phi_{C,D}\Phi^{-1}_{C,D}=\textrm{id}_{\textrm{C}(C,G(D))}$}.
• Show that {$\Phi_{C',D'}\textrm{D}(F(c),d)=\textrm{D}(c,G(d))\Phi_{C,D}$}.
• Show that {$\textrm{D}(F(c),d)\Phi^{-1}_{C,D}=\Phi^{-1}_{C',D'}\textrm{D}(c,G(d))$}.

• Verify the triangle identities.
• Verify the universal mapping property.