Category Theory/Definition, examples

Definition (category):

A category ${\mathcal {C}}$ is a class of objects, together with a class of so-called morphisms, each of which have a domain and a target, and a composition of morphisms, such that the following set of axioms hold, if for any two objects $A$ and $B$ of ${\mathcal {C}}$ the subclass of morphisms with domain $A$ and target $B$ is denoted $\operatorname {Hom} (A,B)$:

1. Whenever either $A\neq A'$ or $B\neq B'$, $\operatorname {Hom} (A,B)$ and $\operatorname {Hom} (A',B')$ are disjoint
2. For any objects $A,B,C$ of ${\mathcal {C}}$ and any morphisms $f:A\to B$ and $g:B\to C$, there exists a morphism $h=g\circ f\in \operatorname {Hom} (A,C)$, called the composition of $f$ and $g$
3. Composition is associative, ie. $f\circ (g\circ h)=(f\circ g)\circ h$
4. Whenever $A$ is an object of ${\mathcal {C}}$, then there exists a unique morphism $1_{A}\in \operatorname {Hom} (A,A)$ that acts as an identity both on the left and on the right for the composition of morphisms.

Exercises

1. If both $f$ and $g$ are set mappings such that $f\circ g$ is injective, prove that $g$ is injective.