On the linear independency of monoidal natural transformations (Q2884414)

Let \(\mathcal C\) be a tensor category over an algebraically closed field \(k\). Let \(\mathcal I\) be a skeletally small monoidal category and let \(F, G: \mathcal{I} \to \mathcal{C}\) be monoidal functors. The author proves that the set of monoidal natural transformations from \(F\) to \(G\) is a linearly independent subset of the \(k\)-vector space of natural transformations from \(F\) to \(G\). As a corollary, if \(\mathcal C\) and \(\mathcal D\) are tensor categories and \(F, G: \mathcal {C} \to \mathcal{D}\) are right exact tensor functors, then the set of monoidal natural transformations from \(F\) to \(G\) is finite if \(\mathcal C\) is finite. In particular, a finite tensor category \(\mathcal C\) has finitely many pivotal structures and the group of monoidal natural automorphisms of the identity functor is finite. Moreover, if \(k\) has characteristic zero and \(S\) is the set of isomorphism classes of simple objects of \(\mathcal C\), then \(Aut_{\otimes}(id_NEWLINE{\mathcal C})\) can be identified with a certain set of functions from \(S\) to \(k^\times\).