The Clowder Project

In detail, a contravariant functor from $\mathcal{C}$ to $\mathcal{D}$ consists of:

1. 1.

   Action on Objects. A map of sets

   \[ F \colon \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright ) \to \operatorname {\mathrm{Obj}}\webleft (\mathcal{D}\webright ), \]

   called the action on objects of $F$.

2. 2.

   Action on Morphisms. For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, a map

   \[ F_{A,B} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Hom}}_{\mathcal{D}}\webleft (F\webleft (B\webright ),F\webleft (A\webright )\webright ), \]

   called the action on morphisms of $F$ at $\webleft (A,B\webright )$.

satisfying the following conditions:

1. 1.

   Preservation of Identities. For each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the diagram

   
   commutes, i.e. we have
   \[ F\webleft (\operatorname {\mathrm{id}}_{A}\webright ) = \operatorname {\mathrm{id}}_{F\webleft (A\webright )}. \]
2. 2.

   Preservation of Composition. For each $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the diagram

   
   commutes, i.e. for each composable pair $\webleft (g,f\webright )$ of morphisms of $\mathcal{C}$, we have
   \[ F\webleft (g\circ f\webright ) = F\webleft (f\webright )\circ F\webleft (g\webright ). \]

Throughout this work we will not use the term “contravariant” functor, speaking instead simply of functors $F\colon \mathcal{C}^{\mathsf{op}}\to \mathcal{D}$. We will usually, however, write

for the action on morphisms

of $F$, as well as write $F\webleft (g\circ f\webright )=F\webleft (f\webright )\circ F\webleft (g\webright )$.