The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity mappings on the old structures, and homomorphisms in this reflective subcategory will preserve the neutral element.
What if we add neutral elements only to semigroups that are not already monoids? Why won't this be a reflective subcategory in $\mathbf{Sem}$? What is the easiest proof of this?