It is well-known that any symmetric monoidal category is equivalent to a strict symmetric monoidal category. The construction of this strict monoidal category is rather technical and it appears to me that there is a significantly easier way of obtaining an even better result. Of course this means almost certainly that I am making a mistake.
Given a symmetric monoidal category $S$, consider the category of isomorphism classes $T$. Explicitly, let $T$ be the full subcategory of $S$ which has one distinct object for every isomorphism class of $S$. The inclusion functor $F:T\to S$ is fully faithful and essentially surjective. Hence it is an equivalence of categories with some inverse $G$.
Now $T$ is strict symmetric monidal with operation $\oplus_T$ defined via
$$A\oplus_T B:=G(F(A)\oplus_SF(B))$$
This is
a) very simple
and
b) also commutativity is strict which appears to be better than the classical result.
So what is going wrong?
Best Answer
There are a couple of issues:
The first thing is that isomorphism classes don't form a category.
You may force them into being a category by picking a representative in each isomophrpism class, and then looking at the full subcategory that they span. But this construction depends on the choice of representative:
If you change your mind, and pick another set of representative, then there is no canonical equivalence of categories between those two full subcategories (there are many equivalences of categories between those two categories, but there is no canonical one).
Ok, maybe that's not too bad after all... This thing that you call "the category of equivalence classes" actually has a name: it's called a skeleton of the category.
Ok, now let's look at another, more serious isue:
A monoidal category (C,⊗) isn't just a category with a ⊗, it's also equipped with an associator α:(X⊗Y)⊗Z→X⊗(Y⊗Z).
If you restrict to a skeleton T⊂C of the category, then in order to equip that skeleton with a monoidal structure, you should pick for every object X of C an isomorphism $$ \beta_X:X\to \underline{X}, $$ where $\underline{X}$ is the chosen representative on the equivalence class $[X]$.
The monoidal product on $T$ is given by $$ X \underline\otimes Y := \underline {X \otimes Y} $$ and it has the following associator:
$$ (X \underline\otimes Y) \underline\otimes Z = \underline{X \otimes Y \otimes Z} = \underline{\underline{X \otimes Y} \otimes Z} $$
$$\xrightarrow{\beta^{-1} _ {\underline{X \otimes Y}\otimes Z}}\underline{X \otimes Y} \otimes Z \xrightarrow{\beta^{-1}_{\underline{X \otimes Y}}\otimes 1}(X \otimes Y) \otimes Z $$
$$\xrightarrow{\alpha} X\otimes (Y \otimes Z) \xrightarrow{1\otimes \beta_{Y \otimes Z}} X \otimes \underline{Y \otimes Z} $$
$$\xrightarrow{\beta_{X \otimes \underline{Y \otimes Z}}} \underline{X \otimes \underline{Y \otimes Z}} = \underline{X \otimes Y \otimes Z} = X \underline\otimes (Y \underline\otimes Z) $$
This associator goes from an object to itself, but is probably not trivial. So the resulting monoidal category is maybe skeletal, but certainly not strict.