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

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

edited Dec 11 at 19:28

asked Dec 11 at 18:21

Auguste Hoang Duc
238●1●1●8

Isn't a well-placed question, is $v$ a funtor or a transformation between funtors $F$ and $G$? – Buschi Sergio Dec 11 at 18:39

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This is a job for... String Diagram Man! :-) – Todd Trimble Dec 11 at 21:04

Nitpick: instead of saying "morphism of functors", you should say "morphism of tensor functors" (there's a distinction). – Todd Trimble Dec 12 at 0:29

Todd Trimble · Accepted Answer · 2012-12-12 00:23:22Z UTC

Okay, here is a link 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 proof, which unfortunately I can't draw for you in a convenient way. The big commutative diagram which results only looks intimidating.

Functors on rigid tensor categories.

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