I have the next doubt about this problem:
"Let $\mathfrak{U}$ be a small category and let $Y:\mathfrak{U}\rightarrow [\mathfrak{U}^{opp},\mathbf{Sets}]$ be the yoneda embedding
$Y(A)=\mathfrak{U}(-,A)$.
Let $J:\mathfrak{U}\rightarrow\mathfrak{B}$ be a functor. Define $R:\mathfrak{B}\rightarrow[\mathfrak{U}^{opp},\mathbf{Sets}]$ on objects by $R(B)=\mathfrak{B}(J-,B)$. Show how to extend this definition to yield a functor $R$, and give reasonable condition under which $Y=RJ$".
I showed that $R$ gives a functor but I do not see the conditions in which $Y=RJ$, beccause I only see that if that condition holds, we have that we have that for any object $A$ in $\mathfrak{U}$ we have that $\mathfrak{U}(C,A)=\mathfrak{B}(JC,JA)$ for all objects in $\mathfrak{U}$.
But I do not see anything to proceed or anything clear for this.
Thank you for your time!
TO be precise, I think the correct way of phrasing your question would be asking that $Y$ and $RJ$ are isomorphic.
It is enough that $J$ is fully faithful: if this is the case, for every pair of objects $A,A'$ of $\mathfrak{U}$, we have $\mathfrak{U}(A',A)\cong \mathfrak{B}(JA',JA)$, which is what you want.