In the book Tensor Categorties by EGNO, there are
Exercise 2.8 Show that any monoidal category $\mathcal{C}$ is monoidally equivalent to a skeletal monoidal category $\bar{\mathcal{C}}$.
In the hint, the authors write
For any two isomorphism clasess $i,j$, fix an isomorphism $\mu_{ij}：X_i\otimes X_j\to X_{ij}$. Let $\bar{\mathcal{C}}$ be the full subcategory of $\mathcal{C}$ consisting of the objects $X_i$, with tensor product defined by $X_i\bar{\otimes} X_j = X_{ij}$, and with all the structure transported using the isomorphisms $μ_{ij}$.Then $\bar{\mathcal{C}}$ is the required skeletal category, monoidally equivalent to $\mathcal{C}$.
I want to know how to check the defined associator satisfies the pentagon axiom in $\bar{\mathcal{C}}$. In all papers and notes I have found, they all omit the specific process.
I will not write out the details too, because it is a good exercise to do it yourself. I think what you might be missing is what it means to have all the structure transported using the isomorphisms $\mu_{ij}$.
In the case of a braiding/symmetry it would mean that the natural isomorphism $\overline\sigma:X_i \overline\otimes X_j \cong X_j \overline\otimes X_i$ is defined via the composite $$X_i \overline\otimes X_j = X_{ij} \overset{\mu_{ij}^{-1}}\rightarrow X_i \otimes X_j \cong_\sigma X_j \otimes X_i \overset{\mu_{ji}}\rightarrow X_{ji} = X_j \overline\otimes X_i$$ Now checking that this braiding is a symmetry (ie that $\overline\sigma\circ\overline\sigma = id$) means checking that the defining composites have this property. You can already see this: if $\sigma$ is a symmetry, then when composing $\overline\sigma$ with itself first the $\mu_{ji}$ of the first will cancel with the $\mu_{ji}^{–1}$ of the second, then the $\sigma$ will cancel itself and then the $\mu_{ij}$ will be killed too.
In the same way you can check out all other axioms. Write out the diagram for $\overline\otimes$, write out the corresponding diagram for $\otimes$, connect them via the maps $\mu_{ij}$ and notice that by construction everything commutes.