- Objects. A class $\text{Obj}\webleft (\mathcal{C}\webright )$ of objects.
- Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, a class $\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$, called the class of morphisms of $\mathcal{C}$ from $A$ to $B$.
- Identities. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map of sets
\[ \mathbb {1}^{\mathcal{C}}_{A}\colon \text{pt}\to \textup{Hom}_{\mathcal{C}}\webleft (A,A\webright ), \]
called the unit map of $\mathcal{C}$ at $A$, determining a morphism
\[ \text{id}_{A} \colon A \to A \]
of $\mathcal{C}$, called the identity morphism of $A$.
- Composition. For each $A,B,C\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map of sets
\[ \circ ^{\mathcal{C}}_{A,B,C} \colon \textup{Hom}_{\mathcal{C}}\webleft (B,C\webright )\times \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{C}}\webleft (A,C\webright ), \]
called the composition map of $\mathcal{C}$ at $\webleft (A,B,C\webright )$.
such that the following conditions are satisfied:
-
Associativity. The diagram commutes, i.e. for each composable triple $\webleft (f,g,h\webright )$ of morphisms of $\mathcal{C}$, we have
\[ \webleft (f\circ g\webright )\circ h = f\circ \webleft (g\circ h\webright ). \]
-
Left Unitality. The diagram
commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have
\[ \text{id}_{B}\circ f=f. \]
-
Right Unitality. The diagram
commutes, i.e. for each morphism $f\colon A\to B$ of $\mathcal{C}$, we have
\[ f\circ \text{id}_{A}=f. \]