Proof of the isomorphism $\textrm{Hom}_{A-mod}(\textrm{Hom}_{A-mod}(A,A),A)\cong A$ for any ring $A$

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Let $A$ be a commutative ring with identity. Then we have an isomorphism $$A \rightarrow \textrm{Hom}_{A-mod}(A,A), a \mapsto (x \mapsto ax)$$, whose inverse is $$\textrm{Hom}_{A-mod}(A,A) \rightarrow A, \varphi \mapsto \varphi(1).$$

Then applying this isomorphism, we get $$\textrm{Hom}_{A-mod}( \textrm{Hom}_{A-mod}(A,A) ,A) \cong \textrm{Hom}_{A-mod}(A,A) \cong A.$$

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