Functors

[!definition] Functor A functor $F=(F_0,F_1)\colon \mathcal{C}\to \mathcal{D}$ constists of maps

$$ F_0\colon \operatorname{D}\to\operatorname{C},\quad F_1\colon \mathcal{C}_1\to\mathcal{D}_1, $$

such that

• if $f\colon A\to B$, then $F_1(f)\colon F_0(A)\to F_0(B)$;
• $F_1(id_A) = id_{F_0(A)}$
• for every composable pair