If there is a specified object $I$, and natural isomorphisms $\lambda_A:\colon A\otimes I\stackrel{\simeq}{\to} A$ and $\rho_A:\colon I\otimes A\stackrel{\simeq}{\to} A$, then it is a magmoidal category with unit. Depending on the intended uses, it might be necessary to also impose the coherence condition that $\rho_I = \lambda_I\colon I\otimes I\stackrel{\simeq}{\to} I$ (which appeared in Mac Lane‘s original definition of a monoidal category, but proved redundant by Max Kelly). Note that the coherence condition usually placed on unitors in monoidal categories cannot even be stated in the context of a magmoidal category, since there is no associator to relate $A\otimes (I\otimes B)$ and $(A\otimes I)\otimes B$.