Base change of equivalences

Exercise 1.3.11. The construction of pullbacks of \(\infty \)-categories is functorial: given a commutative diagram

there is an induced functor \[ \phi \times _{\chi } \psi := (\phi \circ \pr _{C}, \psi \circ \pr _{D})\colon C \times _{E} D \to C' \times _{E'} D'. \] Show that if each of the functors \(\phi \), \(\psi \) and \(\chi \) is an equivalence, then so is \(\phi \times _{\chi } \psi \). Deduce that equivalences are stable under pullback: if \(g\colon D \to E\) is an equivalence and \(f\colon C \to E\) is an arbitrary functor, then the projection \(C \times _{E} D \to C\) is an equivalence.