A category $\mathcal{C}$ consists of the following data:

1. A class of objects, denoted by ${\rm{Ob}}(\mathcal{C})$.
2. To each pair of objects $A, B$, a set ${\rm{Hom}}(A, B)$ of morphisms from $A$ to $B$.
3. To each triple of objects $A, B, C$, a composition law $${\rm{Hom}}(A,B)\times {\rm{Hom}}(B, C)\longrightarrow {\rm{Hom}}(A, C),\ (f,g)\longmapsto f\circ g.$$

Moreover, it subjects to the following axioms:

(1) Composition is associative, i.e. $(f\circ g)\circ h = f\circ (g\circ h)$ for morphisms $f,g,h$.

(2) For each object $A$, there is a unique identity morphism $1_{A}: A\rightarrow A$ such that $1_{A}\circ f=f\circ 1_{A}$ if the composition makes sense.