Functor

Definition

Suppose $\mathcal{C},\mathcal{D}$ are categories. A functor $\mathcal{F}$ from $\mathcal{C}$ to $\mathcal{D}$ comprises the following data:

• A mapping $\mathcal{F}:\mathrm{O}\mathrm{b}\mathcal{C}\to \mathrm{O}\mathrm{b}\mathcal{D}$.
• For any $A,B\in \mathcal{C}$, a mapping $\mathcal{F}:\mathcal{C}(A,B)\to \mathcal{D}(FA,FB)$.

satisfying the following condition:

• For any $A\in \mathcal{C}$, $\mathcal{F}({\mathrm{i}}_{\mathrm{d}})={\mathrm{i}}_{\mathrm{d}}$.
• For any $A,B,C\in \mathcal{C}$, and $f\in \mathcal{C}(A,B),g\in \mathcal{C}(B,C)$, we have $F(g\circ f)=Fg\circ Ff$.