Let $C$ be a Tannakian category (ie. it is rigid tensor Abelian category where hom sets are $k$-vector spaces and there is a fibre functor $w$ from $C$ to category of vector spaces such that $w$ is a exact faithful tensor functor.)
Let group $G$ act on $C$ in the sense that for each $g \in G$ we have a functor $a_g : C \to C$ such that $a_g a_h$ and $a_{gh}$ are naturally isomorphic. Further assume the group acts by tensor functors.
Simpson in his paper "Higgs Bundles and Local Systems" says that "by transport of structure, $G$ also acts on ${\rm End}(w,C)$ where the endomorphism algebra is just the endomorphism of the fibre functor."
My question is : How do we get such an action?
A typical element in ${\rm End}(w,C)$ is a tuple ${f_V}$ where $f_V \in{\rm End}(w(V))$ and makes required diagrams commute. (It's just the data of natural transformation nothing else).