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This is a question about the proof of proposition 1.13 in Deligne and Milne, Tannakian Categories. Let $C,C'$ be two rigid tensor categories and $u F,G : C \rightarrow C'$ a be two tensor functorfunctors. Let $u : F \rightarrow G$ be a morphism of functors. Define the functor morphism $v : G \rightarrow F$ by $$v(X) : G(X) \simeq G(X^\vee)^\vee \xrightarrow{{}^t u(X^\vee)} F(X^\vee)^\vee \simeq F(X).$$

Why is $v$ the inverse of $u$ ?

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# Functors on rigid tensor categories.

This is a question about the proof of proposition 1.13 in Deligne and Milne, Tannakian Categories. Let $C,C'$ be two rigid tensor categories and $u : C \rightarrow C'$ a tensor functor. Define the functor $v : G \rightarrow F$ by $$v(X) : G(X) \simeq G(X^\vee)^\vee \xrightarrow{{}^t u(X^\vee)} F(X^\vee)^\vee \simeq F(X).$$

Why is $v$ the inverse of $u$ ?