Pivotal and spherical structures vary by monoidal automorphisms of the identity

If a tensor category has one pivotal structure, then every monoidal natural automorphism $\eta$ of the identity functor produces another one by $X\stackrel{{\eta }_X}{\to }X\stackrel{a_X}{\to }{X}^{\vee \vee }$. Equivalently, if $a$ and $b$ are pivotal structures, then $a^{-1}b$ is a unique monoidal automorphism of the identity. Thus the set of pivotal structures, when nonempty, is a torsor over ${\mathrm{Aut}}_{\otimes }({\mathrm{id}}_{\mathcal{C}})$. Spherical structures are precisely the members of this set that also satisfy the spherical trace condition; changing by a monoidal automorphism need not preserve that extra condition.

This is why ${\dim }_a(X)$ keeps the subscript $a$: the comparison map is part of the data.