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?
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  • Thank you Philip! It really helps a lot:). I will consider that indeed, it was my first time using this website so I was not sure what to expect

  • I am glad I was able to help.

The answer is accepted.