Category

A category $\mathcal C$ consists of:

1. A collection $\mathrm{Ob}(\mathcal C)$ of objects.
2. For each pair $X,Y \in \mathrm{Ob}(\mathcal C)$, a collection $\mathrm{Hom}_{\mathcal C}(X,Y)$ of morphisms $X \to Y$.
3. For each triple $X,Y,Z$, a composition operation
   $$\circ : \mathrm{Hom}_{\mathcal C}(Y,Z)\times \mathrm{Hom}_{\mathcal C}(X,Y)\to \mathrm{Hom}_{\mathcal C}(X,Z), \quad (g,f)\mapsto g\circ f.$$
4. For each object $X$, an identity morphism $1_X \in \mathrm{Hom}_{\mathcal C}(X,X)$.

These data satisfy:

• Associativity: $h\circ(g\circ f)=(h\circ g)\circ f$ whenever the composites are defined.
• Identity laws: $1_Y\circ f=f=f\circ 1_X$ for every $f:X\to Y$.

These are the category axioms.