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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?

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See Theorem 3.2 in Turning Monoidal Categories into Strict Ones. 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.

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