I am a little puzzled by what people call "the transformed field" in CFT. The usual definition of the scale-invariant function is \begin{equation} \phi(\lambda z) = \lambda^\Delta \phi(z) \end{equation} Clearly, we have same $\phi$ in both LHS and RHS. The natural generalization of this definition is \begin{equation} \phi(f(z)) = \left(\dfrac{\partial f}{\partial z}\right)^\Delta \phi(z) \end{equation} Again, we have same $\phi$ on both sides.
Now, after making the change of variables $z\to f(z)$, (to me) it would make sense to introduce a new function defined as \begin{equation} \phi'(z) \equiv \phi(f(z)) \end{equation}
However, in the standard CFT books I find something completely opposite. Namely, for some reason people put prime in the LHS of the first two equations I wrote.
"Introduction to Conformal Field Theory With Applications to String Theory" by Blumenhagen and Plauschinn:
"Conformal Field Theory" by Di Francesco, Mathieu and Senechal:
I feel like I have a gap in conceptual understanding... Why do they put primes?? Clearly, to define a function $\phi$ having some special property, we need to have it on both sides - just like in the usual definition of the scale-invariant function.