To illustrate, consider the set of vertex colorings of a graph subject to the requirement that adjacent vertices are assigned distinct colors. There is a contravariant functor from the category of graphs to the category of sets that takes a graph to the set of $n$-colorings of its vertices. To explain the contravariance, note that an $n$-coloring of a graph $G$ and a graph homomorphism $G’ \to G$ induce an $n$-coloring of $G’$ that colors each vertex of $G’$ to match the color of its image: as graph homomorphisms preserve the incidence relation on vertices, any two adjacent vertices of $G’$ are assigned distinct colors. This defines the action of the functor $n\text{-}\textup{fun} \colon \textup{\textsf{cat}}^\mathrm{op} \to \textup{\textsf{cat}}$ on morphisms.