Functor

A functor is the correct notion of homomorphism between Categories. Explicitly, a functor $F:\mathcal{C}\to\mathcal{D}$ between two categories consists of

• A mapping of objects $F_{0}:\mathcal{C}_{0}\to\mathcal{D}_{0}$
• A mapping of morphisms $F_{1}:{{X}\to{Y}}\to{{F_{0}(X)}\to{F_{0}}}(Y)$, such that
• $F_{1}(mathrm{id}{ifx..lse\par i})=mathrm{id}{ifx..lse\par i}$
• $F_{1}(f\circ g)=F_{1}(f)\circ F_{1}(g)$