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Here is a cute observation: Let $F,G : \mathcal{C} \to \mathcal{D}$ be a symmetric monoidal functors between symmetric monoidal categories, and let $\eta : F \to G$ be a monoidal transformation. Then for every dualizable object $V \in \mathcal{C}$ the morphism $\eta_V : F(V) \to G(V)$ is actually an isomorphism! The inverse is given by

$G(V) \cong G(V^*)^* \xrightarrow{\eta_{V^*}*} F(V^*)^* \cong F(V).$

Of course one has to check that this is, in fact, inverse to $\eta_V$. A nice corollary: If $F,G$ are cocontinuous, and every object of $\mathcal{C}$ is a colimit of dualizable objects (which happens quite often), then every monoidal transformation $F \to G$ is an isomorphism, i.e. $\mathrm{Hom}_{c\otimes}(F,G)$ is a groupoid.

Is this well-known? Does somebody know a reference for this observation? The proof is not so hard, but this is one of dozens of lemmas which I need in my thesis and with which I don't want to waste time / place with the proof. In an example this appears in Remark 2.4.7 in Lurie's On the classification of topological field theories.

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This goes back at least to Saavedra-Rivano "Categories Tannakiennes." In fact, this has a generalisation to Frobenius functors, in Day-Pastro "Note on Frobenius monoidal functors," and to more general settings, as monoidal bicategories: Lopez Franco, Street, Wood; "Duals Invert," (App. Categor. Str. Vol 19).

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    $\begingroup$ Thank you. In Categories Tannakiennes this is noted (with a sketchy proof, though) in the special case of rigid $\otimes$-categories in Chapter I, Proposition 5.2.3. In Note on Frobenius monoidal functors it is Proposition 7. In Duals Invert, it is Theorem 3.2. $\endgroup$ – Martin Brandenburg Sep 13 '13 at 8:40

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