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.

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.