How to show that the composition of two riemannian isometries is an isometry?

If for Riemannian charts $(P, g)$, $(Q, h)$ and $(R, i)$ I have two Riemannian isometries (differentiable bijection with differentiable inverse), one $\phi: (P, g) \to (Q,h)$ such that $g = \phi^{*}h$ and one $\rho: (Q,h) \to (R, i)$ such that $h = \rho^{*}i$, how do I show that the composition of these two isometries is again an isometry?
See the file/image for more information on the used notation here.