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6 Sample mathematics
6.1 Adjoint functors and coadjoint functors via universal arrows
Let \(\Cc \) and \(\Dc \) be categories. Let \(G\colon \Dc \to \Cc \) be a functor.
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Definition 1. Let \(A\) be an object of \(\Cc \). A universal arrow from \(A\) to \(G\colon \Dc \to \Cc \) is a pair:
\[ \left (F_A, A \ra {\eta _A} G(F_A)\right ), \]
where \(F_A\) is an object in \(\Dc \) and \(\eta _A\colon A \to G(F_A)\) is an arrow in \(\Cc \), such that the following universal property is satisfied:
For any object of \(B\) of \(\Dc \) and any arrow \(f\colon A \to G(B)\), in \(\Cc \), there exists a unique arrow \(\hat {f}\colon F_A \to B\), in \(\Dc \), that makes the following diagram, in \(\Cc \), commute:
\[\xymatrix {& A \ar [drr]_f\ar [rr]^{\eta _A} && G(F_A) \ar @{-->}[d]^{G(\hat {f})}\\ &&& G(B)} \]
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Exercise 2. In the conditions of the previous definition, prove that if \(\left (F_A, A \ra {\eta _A} G(F_A)\right )\) is a universal arrow from \(A\) to \(G\), then we have a bijection:
\[\phi _{A,B}\colon \hom _\Cc \big (A,G(B)\big ) \longrightarrow \hom _\Dc (F_A,B), \]
such that
\[ \big (f\colon A \to G(B)\big ) \stackrel {\phi _{(A,B)}}{\longmapsto } (\hat {f}\colon F_A \to B\big ).\]
Moreover, prove that the bijection \(\phi _{A,B}\) is natural in \(B\). This means that given any arrow \(g\colon B \to C\), in \(\Dc \), the following diagram (in the category of sets) commutes:
\[ \xymatrix { & \hom _\Cc (A,G(B)) \ar [d]_{m\mapsto G(g)\circ m } \ar [rrrr]^{\phi _{A,B}} &&&& \hom _\Dc (F_A,B) \ar [d]^{n\mapsto g\circ n } \\ & \hom _\Cc (A,G(C)) \ar [rrrr]_{\phi _{A,C}} &&&& \hom _\Dc (F_A,C)\\ } \]