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
\[ F_{A,B}\colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )\to \textup{Hom}_{\mathcal{D}}\webleft (F\webleft (B\webright ),F\webleft (A\webright )\webright ) \]
for the action on morphisms
\[ F_{A,B}\colon \textup{Hom}_{\mathcal{C}^{\mathsf{op}}}\webleft (A,B\webright )\to \textup{Hom}_{\mathcal{D}}\webleft (F\webleft (A\webright ),F\webleft (B\webright )\webright ) \]
of $F$, as well as write $F\webleft (g\circ f\webright )=F\webleft (f\webright )\circ F\webleft (g\webright )$.
