I was wondering if we needed both rules – "\$$f\$$ preserves \$$\otimes\$$" and "\$$f\$$ preserves \$$I\$$" – in the definition of a monoid homomorphism.

Or is "\$$f\$$ preserves \$$\otimes\$$" enough on its own?

If we can pick an \$$x\$$ such that \$$f(x) = I_Y\$$ and then we have \$f(I_X) = f(I_X) \otimes I_Y = f(I_X) \otimes f(x) = f(I_X \otimes x) = f(x) = I_Y\$

But what if \$$I_Y\$$ is *not* in the image of \$$f\$$?

I suspect we might be able to get a counterexample in this case – but I'm struggling to come up with one.