Right adjoint to the forgetful functor $text{Ob}$

Let $text{Ob}:textbf{Cat}rightarrowtextbf{Set}$ be the forgetful functor mapping a small category to its set of objects. Consider the functor $R:mathbf{Set}rightarrowtextbf{Cat}$ 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 $x,yin X$, 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?