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Proof of Yoneda's lemma in category theory.

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\begin{diagram}[tight]
\Hom(A,A) & & \rTo^{\Hom(A,f)} & & \Hom(A,X) \\
 & \scriptstyle\id_A & \rMapsTo & \scriptstyle f & \\
\dTo<{\Phi_A} & \dMapsTo  & & \dMapsTo & \dTo>{\Phi_X} \\
 & \scriptstyle u & \rMapsTo & \scriptstyle (Ff)u = \Phi_X(f) & \\
F(A) & & \rTo_{Ff} & & F(X) \\
\end{diagram}