Given two monoids we always have a morphism from one to the other thanks to the presence of the identity element.
Are there examples of non-empty semigroups that have no morphisms from one to the other? The destination semigroup can't be finite, because if we have an idempotent present, we just map everything to it and get a constant morphism. Apparently, it can't contain an idempotent, period.
Put another way, the subcategory of finite semigroups is clearly strongly connected. Is the same true of the category of all semigroups?
Are there two such non-empty semigroups that don't have a morphism in either direction?