Some examples of morphisms are homomorphisms from the categories studied in universal algebra (such as those of groups, rings, and so on), continuous functions between topological spaces, elements of a group when the group is thought of a special kind of category, paths in a single topological space (which form a groupoid), functors between categories, and many more.

Variants and subclasses of morphism:

- Every object
*X*in every category has an*identity morphism***id**_{X}which acts as an identity under the operation of composition. - If
*f*:*X*->*Y*and*g*:*Y*->*X*satisfy*f*o*g*=**id**_{Y}, then*f*is a retraction and*g*is a section. - If
*f*is both a retraction*and*a section, then it is an isomorphism. In this case, the objects*X*and*Y*should be thought of as completely equivalent for purposes of the category*C*. - A morphism
*f*:*X*->*X*is an endomorphism of*X*. - An endomorphism that is also an isomorphism is an automorphism.
- Suppose that whenever
*g*:*Y*->*Z*and*h*:*Y*->*Z*and*g*o*f*=*h*o*f*, it always turns out that*g*=*h*. Then*f*is an epimorphism. Every retraction must be an epimorphism. It's also called an epi or an epic.

- Suppose that whenever
*g*:*W*->*X*and*h*:*W*->*X*and*f*o*g*=*f*o*h*, it always turns out that*g*=*h*. Then*f*is a monomorphism. Every section must be a monomorphism. It's also called a mono or a monic.

- If
*f*is both an epimorphism*and*a monomorphism, then*f*is a bimorphism. Note that not every bimorphism is an isomorphism! However, any morphism that is both an epimorphism and a section, or both a monomorphism and a retraction, must be an isomorphism. - A homeomorphism is simply an isomorphism in the category of topological spaces.
- A diffeomorphism is simply an isomorphism in the category of differentiable manifolds.\n