In a cartesian monoidal category we have the product with two projections $\pi_1$ and $\pi_2$, and the terminal object $1$. We also have unitors $\rho_A \colon A \times 1 \to A$ and $\lambda_A \colon 1 \times A \to A$. The obvious unitors are given by projections: $\rho = \pi_1$ and $\lambda = \pi_2$. Is it possible to have, for the same product and terminal object, a different set of unitors (and, by analogy, associators)? Would it still be called a cartesian monoidal category?

Note: Unitors determine projections. Given $! \colon A \to 1$ the unique morphism to the terminal object, we can define $\pi'_1: A \times B \to A$ as $\pi'_1 = \rho \circ (A \times \, !)$. But is this the same projection that defines the product?

specificproduct object $A \otimes B$ - which you must, because $\otimes$ has to be a functor - then you are asking if that object can be a product in more than one way. That is, can the object have multiple sets of valid projections. Is that right? (My problem with your wording is when you say "is it possible to havefor the same producta different set of unitors - when you fix the product, you fix $\pi_1$.) $\endgroup$ – statusfailed Oct 30 '19 at 21:11