In detail, a contravariant functor from $\mathcal{C}$ to $\mathcal{D}$ consists of:
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Action on Objects. A map of sets
\[ F \colon \text{Obj}\webleft (\mathcal{C}\webright ) \to \text{Obj}\webleft (\mathcal{D}\webright ), \]
called the action on objects of $F$.
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Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map
\[ F_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{D}}\webleft (F\webleft (B\webright ),F\webleft (A\webright )\webright ), \]
called the action on morphisms of $F$ at $\webleft (A,B\webright )$.
satisfying the following conditions:
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Preservation of Identities. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram
commutes, i.e. we have
\[ F\webleft (\text{id}_{A}\webright ) = \text{id}_{F\webleft (A\webright )}. \] -
Preservation of Composition. For each $A,B,C\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram commutes, i.e. for each composable pair $\webleft (g,f\webright )$ of morphisms of $\mathcal{C}$, we have
\[ F\webleft (g\circ f\webright ) = F\webleft (f\webright )\circ F\webleft (g\webright ). \]
