Examples of mapping two sets where if the two sets are isomorphic doesn't imply that mapping is also 1 to 1 I am struggling with getting an example of two sets S and T and (onto) mapping f, where the fact S and T are isomorphic does not imply that f is also 1 - 1. If possible could you also give an example in which the fact that they are isomorphic would imply that they are 1 - 1?
Thank You!

Consider the map $f: \mathbb{N} \to \mathbb{N}$ given by
$$\begin{aligned} 1 &\mapsto 1,\\
                  2 &\mapsto 1,\\
                  3 &\mapsto 2,\\
                  4 &\mapsto 3, \\
                  5 &\mapsto 4,\\
                     & \ \ \vdots\end{aligned}$$
ie 
$$f(n) = \begin{cases}
                        1 &\text{ if } n =1,\\
                        n - 1 &\text{ otherwise.} \end{cases} $$
