The Clowder Project

Let $(X,x_0)$ and $(Y,y_0)$ be pointed sets and let $f,g,h:(X,x_0)\to (Y,y_0)$ be morphisms of pointed sets.

1. 1. Associativity. We have isomorphisms of pointed sets
   $$\underset{=\text{CoEq}(\text{coeq}(f,g)\circ g,\text{coeq}(f,g)\circ h)}{\underbrace{\text{CoEq}(\text{coeq}(f,g)\circ f,\text{coeq}(f,g)\circ h)}}\cong \text{CoEq}(f,g,h)\cong \underset{=\text{CoEq}(\text{coeq}(g,h)\circ f,\text{coeq}(g,h)\circ h)}{\underbrace{\text{CoEq}(\text{coeq}(g,h)\circ f,\text{coeq}(g,h)\circ g)}},$$

   where $\text{CoEq}(f,g,h)$ is the colimit of the diagram
   

   in ${\mathsf{Sets}}_{\ast }$.

2. 2. Unitality. We have an isomorphism of pointed sets
   $$\text{CoEq}(f,f)\cong B.$$
3. 3. Commutativity. We have an isomorphism of pointed sets
   $$\text{CoEq}(f,g)\cong \text{CoEq}(g,f).$$