Definition 4.2.8. A *functor* $F : \mathcal{A} \to \mathcal{B}$ between two categories $\mathcal{A}, \mathcal{B}$ is given by the following data:

A map $F : \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$.

For every $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ a map $F : \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(F(x), F(y))$, denoted $\phi \mapsto F(\phi )$.

These data should be compatible with composition and identity morphisms in the following manner: $F(\phi \circ \psi ) = F(\phi ) \circ F(\psi )$ for a composable pair $(\phi , \psi )$ of morphisms of $\mathcal{A}$ and $F(\text{id}_ x) = \text{id}_{F(x)}$.

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