Morphism

A morphism in Algebra is bascailly changing the coordinates of a vector using the same space or going from one space to another. Here is a list of some morphism:

• Monomorphism: f: X -> Y is a monomorphism if f ∘ g1 = f ∘ g2 .It implies that g1=g2 and X ->Z
• Epimorphism: f:X -> Y is a epimorphism if f ∘ g1 = g ∘ g2 implying that g1 = g2 and Y -> Z

       A monomorphism and an epimorphism together form a bimorphism.

• Isomorphism: f: X -> Y is an isomorphism if a morphism g : Y → X such that f ∘ g = id_Y and g ∘f = id_X. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f.
• Endomorphism:  f : X → X is an endomorphism of X. A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h ∘ g with g ∘ h = id. 

       An isomorphism and an endomorphism from a automorphism.