Let $\text{Ob}:\textbf{Cat}\rightarrow\textbf{Set}$ be the forgetful functor mapping a small category to its set of objects. Consider the functor $R:\mathbf{Set}\rightarrow\textbf{Cat}$ mapping a set $X$ to the category having $X$ as its set of objects and a single morphism between each pair of objects. I am trying to show that $R$ is right adjoint to $\text{Ob}$.
For $x,y\in X$, should the single morphism between $x$ and $y$ in $R(X)$ be unique? Or, can I use the same morphism for each pair of objects in $R(X)$?–Does it matter?
Edit:
Giving unique arrows would complicate the matter because I would have to specify the composition for each pair; whereas, in giving a constant morphism $*$, I can merely specify $*\circ*:=*$. Right?