The inverse Gaussian distribution $IG(\mu,\lambda)$ is associated with the density $$f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\,\exp\left\{-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right\}\qquad \lambda,\mu,x>0$$ In Schuster (1968), the following connection with the $\chi^2(1)$ distribution is made: if $X\sim IG(\mu,\lambda)$ then$$Z=\frac{\lambda(X-\mu)^2}{2\mu^2X}\sim\chi^2(1)$$ When looking at the proof

I cannot fill the gap between the definition of $Z$ [as a one-to-one transform of $Y$] and the "immediate" conclusion that it is a $\chi^2(1)$ variate. The 1978 review by Folks and Chhikara does not provide further enlightenment.