In section A.26 page 496 here (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), it is written:
A pair $(A,a), a \in F(A)$ is said to represent $F$ if the natural transformation $$T_a: h^A \rightarrow F, (T_a)_R(f) = F(f)(a)$$ is an isomorphism.
I can't seem to make sense out of the notation. I don't know where the natural transformation $T_a$ is coming from or what the definition means. What should this be, if it is not correct?
So, $h^A$ is the contravariant $\mathrm{Hom}$ functor associated to the object $A$. It sends an object $B$ to the set $\mathrm{Hom}(B,A)$. A natural transformation $T$ from $h^A$ to another contravariant functor $F$ (on whatever category $A$ comes from) is a collection of functions $h^A(B)=\mathrm{Hom}(B,A)\to F(B)$ for each object $B$, compatible with functoriality of $h^A$ and $F$ in the obvious sense. In this case, for an element $a\in F(A)$, Milne is defining $T_{a,B}:h^A(B)\to F(B)$ by the sending $f:B\to A$ to $F(f)(a)$. Note that $F(f):F(A)\to F(B)$, so this definition makes sense. Checking that the collection of maps $T_{a,B}$ for varying $B$ constitute a natural transformation $T_a:h^A\to F$ is an exercise.
(Note: I don't have Milne's notes in front of me, and I'm assuming $h^A$ is contravariant, as this is more common, but the same type of thing works for a covariant functor $F$ if one uses the covariant $\mathrm{Hom}$ functor $\mathrm{Hom}(A,-)$.)