Trying to define a functor with exponentials

Question

I need to define a functor $F:C \rightarrow C$ that maps $X$ to $R^X$, where $R$ is a fixed object. The thing is, given an arrow $f:X \rightarrow Y$, $F(f)$ has to be an arrow from $R^X$ to $R^Y$, but I can't find the adequate one.

Can anyone help me with this? Thanks.

Answer 1

$F$ won't be covariant, but rather a contravariant functor. If $X \to Y$ is a morphism, this induces a natural transformation $- \times X \to - \times Y$, which in turn induces a natural transformation $\hom(-,R^Y) \cong \hom(- \times Y,R) \to \hom(- \times X,R) \cong \hom(-,R^X)$, i.e. a morphism $R^Y \to R^X$.