To be clear, we define a new process from the original Wiener process Wt.
It can then be shown that that new process (call it Wt_bar) is itself a Wiener process.
I'd caution on equating the two processes as such, since the original process W_t is a stochastic process, it is not "guaranteed" that it's value at t will meet the above definition - i.e. W(t) need not equal t*W(1/t) - i.e. the value it realised at 1/t. However, in the probabilistic sense this is correct and thus to not confuse between realised values and the process, I'd keep the Wt_bar to show the new process,since the statement has said that Wt_bar is a Wiener process.
In fact, the following processes can be shown to be SBM (or Wiener):