| context | $\mathcal{O},M$ … set |
| definiendum | $ \langle \mathcal{O},M,id,* \rangle \in \mathrm{it}$ |
| definition | $\mathrm{Mor}:\mathcal{O}\times\mathcal{O}\to M$ |
| definition | $\circ:{\large\prod}_{A,B,C:\mathcal{O}}\,\mathrm{Mor}(B,C)\times\mathrm{Mor}(A,B)\to\mathrm{Mor}(A,C)$ |
| definition | $id:{\large\prod}_{A:\mathcal{O}}\,\mathrm{Mor}_O(A,A)$ |
| postulate | $\mathrm{Mor}(A,B)\cap\mathrm{Mor}(U,V)\ne\emptyset\implies U=A\land V=B$ |
| postulate | $(g\circ f)\circ h=g\circ (f\circ h)$ |
| postulate | $f\circ id_A=id_A\circ f=f$ |
Within set theory, we can define a category as quintuple given by two sets and two (polymorphic) maps into them.
The three axioms say the following: The hom-sets are pairwise disjoint, the composition is associative and $id$ denotes the identity.