Is the condition:
$f(x_1)\otimes_Y f(x_2) \le_Y f(x_1') \otimes_Y f(x_2') \textrm{ implies } f(x_1 \otimes_X x_2) \le_Y f(x_1' \otimes_X x_2')$
really necessary? I'm having a hard time trying to prove it. What I mean in detail is: Assuming that \$$(X,\le_X,\otimes_X,1_X) \$$ is a monoidal preorder, prove that the condition must hold.