The Clowder Project

A category $(\mathcal{C},{\circ }^{\mathcal{C}},{\mathbb{1}}^{\mathcal{C}})$ consists of:

• Objects. A class $\text{Obj}\left(\mathcal{C}\right)$ of objects.
• Morphisms. For each $A,B\in \text{Obj}\left(\mathcal{C}\right)$, a class ${\text{Hom}}_{\mathcal{C}}(A,B)$, called the class of morphisms of $\mathcal{C}$ from $A$ to $B$.
• Identities. For each $A\in \text{Obj}\left(\mathcal{C}\right)$, a map of sets
  $${\mathbb{1}}_A^{\mathcal{C}}:\text{pt}\to {\text{Hom}}_{\mathcal{C}}(A,A),$$

  called the unit map of $\mathcal{C}$ at $A$, determining a morphism

  $${\text{id}}_A:A\to A$$

  of $\mathcal{C}$, called the identity morphism of $A$.

• Composition. For each $A,B,C\in \text{Obj}\left(\mathcal{C}\right)$, a map of sets
  $${\circ }_{A,B,C}^{\mathcal{C}}:{\text{Hom}}_{\mathcal{C}}(B,C)\times {\text{Hom}}_{\mathcal{C}}(A,B)\to {\text{Hom}}_{\mathcal{C}}(A,C),$$

  called the composition map of $\mathcal{C}$ at $(A,B,C)$.

1. 5. Associativity. The diagram
   
   commutes, i.e. for each composable triple $(f,g,h)$ of morphisms of $\mathcal{C}$, we have
   $$(f\circ g)\circ h=f\circ (g\circ h).$$
2. 6. Left Unitality. The diagram
   

   commutes, i.e. for each morphism $f:A\to B$ of $\mathcal{C}$, we have

   $${\text{id}}_B\circ f=f.$$
3. 7. Right Unitality. The diagram
   

   commutes, i.e. for each morphism $f:A\to B$ of $\mathcal{C}$, we have

   $$f\circ {\text{id}}_A=f.$$