I'm working on the following exercise from Chapter 2 of Hilton/Stammbach's A Course in Homological Algebra:
"Let $\mathfrak{A}$ be a small category and let $Y : \mathfrak{A} \to [\mathfrak{A}^\text{opp},\mathfrak{S}]$ be the Yoneda embedding $Y(A) = \mathfrak{A}(-,A)$. Let $J : \mathfrak{A} \to \mathfrak{B}$ be a functor. Define $R : \mathfrak{B} \to [\mathfrak{A}^\text{opp},\mathfrak{S}]$ on objects by $R(B) = \mathfrak{B}(J-,B)$. Show how to extend this definition to yield a functor $R$, and give reasonable conditions under which $Y = RJ$."
I was able to fully define the functor $R$, but I'm confused about the last part of the problem. For $A \in \mathfrak{A}$, $Y(A) = \mathfrak{A}(-,A)$ and $RJ(A) = \mathfrak{B}(J-,JA)$. For these two functors to be equal, we must have $\mathfrak{A}(X,A) = \mathfrak{B}(JX,JA)$ for all $X \in \mathfrak{A}$. But these are morphism sets in two different categories, and so discussing whether or not they're equal (as sets) is more about the exact formal way the categories are defined: You could have two different categories $\mathfrak{B}_1$ and $\mathfrak{B}_2$ which are exactly the same except for the literal way their morphism sets are defined, and with $Y = RJ$ for the first but not for the second. I'd be surprised if this was the point of the exercise. It seems more likely that the authors are asking whether or not $Y$ and $RJ$ are isomorphic in $[\mathfrak{A}^\text{opp}, \mathfrak{S}]$, i.e. equivalent functors; and not whether they are literally equal. Is this true?
Functors are almost never equal. What is more likely to happen is that they are isomorphic, and this is what is meant here, as you have already guessed. More precisely, there is a canonical morphism $Y \to RJ$, and the exercise asks when that morphism is an isomorphism. Which is of course equivalent to $J$ being fully faithful.