Isomorphic objects have the same dimension (pivotal categories)

I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e.,$$\mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \mathrm{Tr}^{L}(\mathrm{id}_{Y}) = \mathrm{dim}(Y),$$where $\mathrm{Tr}^{L}$ is the left quantum trace (see Tensor Categories-EGNO Ch 4.7). Hence, I want to show that the following diagram, corresponding to the dimensions of $X$ and $Y$, commutes:

Here $f\colon X \to Y$ is an isomorphism and $f^{\ast-1}$ is the dual morphism of $f^{-1}$.

I was able to show that the inner square commutes, but I am having trouble showing that the outer triangles commute. I tried taking the explicit form of $(f^{-1})^{\ast}$ but I don't get that the composition with $\mathrm{coev}_{X}$ should be $\mathrm{coev}_{Y}$. I would appreciate a hint for this last part.