Every tricategory is equivalent to a Gray-categories. However any Gray-category is not equivalent to a 3-category. As far as I know, this is similar to the fact that braided monoidal categories are not equivalent to symmetric monoidal categories. However the difference between braided and monoidal categories lies in just one axiom: the fact that $B_{x,y} \star B_{y,x}$ is the identity.

So my question is: could you add such an axiom to tricategories in order to make them 'strictifyable' into 3-categories? And if the answer is yes, what isn't it a part of the definition of a tricategory?

I think such an axiom may look something like $$\omega_{f,g} \star_2 \omega_{g,f} = 1_{f \star_1g}$$ whenever the left-hand-side term is defined, that is when $f$ and $g$ have the same identity as sources and targets.

Side note: of course one could also add this axiom to Gray categories. However I think it cannot be directly encodded into the Gray tensor product, making the definition a bit awkward.

braidedmonoidal category is a (weak) 3-category with one object and one morphism. A symmetric monoidal category is an object like this satisfying an additional constraint, one description of which is that it's deloopable to a (weak) 4-category with one object, one morphism, and one 2-morphism. (This is special to the low category number we're working with: in general being symmetric is a structure on top of being braided, not a property.) $\endgroup$ – Qiaochu Yuan Feb 25 '15 at 19:311more comment