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 $F,G : C \rightarrow C'$ be two tensor functors. Let $u : F \rightarrow G$ be a morphism of functors. Define the morphism $v : G \rightarrow F$ by
$$ v(X) : G(X) \simeq G(X^\vee)^\vee \xrightarrow{{}^t u(X^\vee)} F(X^\vee)^\vee \simeq F(X).$$



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> Why is $v$ the inverse of $u$ ?
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Functors on rigid tensor categories.
Okay, here is a [link](http://ncatlab.org/toddtrimble/published/Morphisms+between+tensor+functors) to my web at the nLab which provides a diagrammatic proof (for one of the two equations that must be verified; the other equation is established similarly). 


Of course, the gigantic diagram which you will find did not spring from my head like Pallas Athena. It was assembled by first studying a simple [string diagram](http://ncatlab.org/nlab/show/string+diagram) proof, which unfortunately I can't draw for you in a convenient way. The big commutative diagram which results only *looks* intimidating. 


