The more common morphisms are:
1. Homomorphism (Similarity between 2 different structures) 同态
Analogy: Similar triangles of 2 different triangles.
2. Isomorphism (Sameness between 2 different structures) 同构
Analogy: Congruence of 2 different triangles
Example: 2 objects are identical up to an isomorphism.
3. Endomorphism (Similar structure of self) = {Self + Homomorphism} 自同态
Analogy: A triangle and its image in a magnifying glass.
4. Automorphism (Sameness structure of self) = {Self + Isomorphism} 自同构
Analogy: A triangle and its image in a mirror; or
A triangle and its rotated (clock-wise or anti-clock-wise), or reflected (flip-over) self.
5. Monomorphism 单同态 = Injective + Homomorphism 
6. Epimorphism 满同态 = Surjective + Homomorphism
Mathtuition88 