I believe that every skeletal monoidal category is monoidally equivalent to a skeletal monoidal category with strict units. Does anybody know a reference for this fact in the literature?



Skeletal monoidal categories with strict units
See Theorem 3.2 in [Turning Monoidal Categories into Strict Ones](http://nyjm.albany.edu/j/2001/7-16p.pdf).
Thus, any monoidal category $(\mathcal{C},\otimes, I,\alpha, \lambda,\rho)$ is monoidally equivalent to a monoidal category $(\mathcal{C},\otimes', I,\alpha')$ with strict unit (note that $\mathcal{C}$ is the same underlaing category). The case in that $\mathcal{C}$ is skeletal, answers your question.


