Is the canonical morphism from $A$ to $R^{R^A}$ monic? Let $\mathcal{C}$ be a distributive category (with finite products and finite coproducts) and let $R$ be an object of $\mathcal{C}$ such that for any object $A$ of $\mathcal{C}$ the exponential $R^A$ exists in $\mathcal{C}$. Let $\partial_A \colon A\to R^{R^A}$ be the morphism obtained by currying the evaluation morphism from $A\times R^A$ to $R$. Is $\partial_A$ monic? Are there some assumptions about $R$ which would make $\partial_A$ a monomorphism?

$\partial_A$ is not always monic. Let $\mathcal C$ be the category of sets and let $R$ be a one-element set.  Then $R^A$ exists and is a one-element set for all $A$, so $\partial_A$ won't be monic when $A$ has 2 or more elements.  
I don't immediately see nice conditions to make $\partial_A$ monic, though in the category of sets it would suffice to require $R$ to have at least two elements.  Intuitively, you want that distinct points in $A$ can be separated by maps to $R$.
