Definition 1.4 (Functor). Let $C$ and $D$ be categories. A functor $F:C\to D$ consists of mappings $F:\mathrm{ob}C\to \mathrm{ob}D$ and $F+\mathrm{mor}C\to \mathrm{mor}D$ such that:

• $F(\mathrm{dom}f)=\mathrm{dom}Ff$

• $F(\mathrm{cod}f)=\mathrm{cod}Ff$

• $F(1_A)=1_{FA}$

• $F(fg)=(Ff)(Fg)$ whenever $fg$ is defined.

We write $\mathbf{\text{Cat}}$ for the category of small categories and the functors between them.