Suppose we have a monoidal category $(C,\otimes,I)$ with left and right unitor being $\lambda$ and $\rho$. They yield two morphisms $\lambda_I,\rho_I:I\otimes I\to I$. It seems to me that both morphisms should always be identical, but I am stuck on finding a way to prove this.
2026-03-25 21:49:32.1774475372
Square of unit in a monoidal category
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As varkor points out, Kock's Elementary remarks on units in monoidal categories gives a concise proof in Lemma 1.5.
Another reference is Corollary 2.2.5 in Tensor categories by Etingof, Gelaki, Nikshych, Ostrik.