For F: X x I-->Y, defined by F(x,t) = y, next define G: Y x I-->X by G(y,u) = x. Then for t = u, we have
F[G(y,t),t] = F{G[F(x,t),t]}, which will ideally be ##\mathbb{1}##. Given Hatcher's definitions pp. 2-3, to me it's not clear how to "invert" a homotopy without an inverse function--let...