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?