If $\displaystyle f:X->Y$ is bijective and $\displaystyle g$ is any map s.t $\displaystyle g:X->X $

then $\displaystyle f \;o \; g \;o f^{-1}$ is bijective.

the compostion is a function that goes from Y to Y. I am pretty sure it's onto, since f is onto. It's injective because g uniquely determines what the composition ends up as.

(we are sending $\displaystyle g:X-X$ to a unique function, that goes from $\displaystyle Y->Y)$

I think the real question I'm asking is, does this define an isomorphism?

Sorry if this is unclear. I tried my best to word it coherently. I will try to clarify if needed.