(Manifolds) Prove the following is a sufficient condition for a function to be a morphism between manifolds

I was hoping for a hint for the following question. I though that I need to use property 3 of the definition of Atlas for to prove it (about two charts with non empty intersection) but cannot progress further. I would appreciate any hint, just not an answer please I want to solve it for my self ( it is a problem for self exercise) 🙂

The problem:

Show that it is sufficient for a map $f : M rightarrow N$ to be a morphism of smooth manifolds if the following holds: for every $x in M$ there exists a chart $left(U, phiright) , x in U$, and there exists a chart $left(V, psiright) , f(x) in V$ , with $psi circ f circ phi^{-1}$ being an infinitely differentiable map.

Thank you !