On the linear independency of monoidal natural transformations

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Let F F , G : I → C G: \mathcal {I} \to \mathcal {C} be monoidal functors from a monoidal category I \mathcal {I} to a linear abelian rigid monoidal category C \mathcal {C} over an algebraically closed field k \mathbf {k} . Then the set N a t ( F , G ) \mathrm {Nat}(F, G) of natural transformations F → G F \to G is naturally a vector space over k \mathbf {k} . Under certain assumptions, we show that the set of monoidal natural transformations F → G F \to G is linearly independent as a subset of N a t ( F , G ) \mathrm {Nat}(F, G) . As a corollary, we can show that the group of monoidal natural automorphisms on the identity functor on a finite tensor category is finite. We can also show that the set of pivotal structures on a finite tensor category is finite.