@Matthew and @Anindya

I follow that making one of the elements the identity of the monoid leads to a contradiction but having hard time thinking about it intuitively. It seems like if you make \$$a=I\$$ then c=d and the poset collapses into a preorder with no monoidal product since \$$(c \otimes d) \leq (a \otimes I) = c \leq a\$$. Is this right?

What about the poset is making it so that it can't have a monodical product? Can we add or remove elements or arrows to make it possible?

Sorry for being slow...