From my limited exposure to the term "natural equivalence", I form the idea that "natural equivalence" is a fanciful way of saying "bijection" (?!).
For instance, in the above text (from Switzer's Algebraic Topology Book), by looking at the proof, it seems that the proof of natural equivalence is just proving one-one correspondence (i.e. bijection).
I am quite sure that "natural equivalence"="bijection" is not correct, so how do I interpret it correctly?
Thanks!
It is possible that indeed the authors are using the term natural just to say that the bijection is the obvious one, meaning that they are using the word natural in a not-categorical way.
On the other hand that bijection is actually natural in the sense of category theory: that is the mappings $(i_X \colon [*,X] \to \pi_0(X))_X$ make commute the diagrams $$\require{AMScd} \begin{CD} [*,X] @>{i_X}>> \pi_0(X) \\ @V[*,f]VV @VV\pi_0(f)V \\ [*,Y] @>>{i_Y}> \pi_0(Y) \end{CD}$$ where $f \colon X \to Y$ is a continuous map between topological spaces, $[*,f]$ and $\pi_0(f)$ are the images of $f$ through the functors $[*,-],\pi_0 \colon \mathbf{Top} \to \mathbf{Set}$.
The proof that the $i_X$'s form a natural equivalence it is a matter of simple calculations (i.e. verifying the commutativity of the diagrams as above).