Fundamental theorem on homomorphisms for structures with only identity element

Question

Is there a version of the fundamental theorem on homomorphisms for binary structures that only have a unique identity element? I'm guessing that there isn't because without the cancelation laws I can't find a way to show that a homomorphism preserves the identity, so there is no sence in defining the kernel. If that's correct, is there a version of the theorem for structures with an identity and with the cancelation laws? In general, what is the most general structure for which the theorem holds? Thanks in advance!