Pullback Connection is Connection on Smooth Manifold

Theorem
Let $M$ and $\tilde M$ be smooth manifolds with or without boundaries.

Let $\phi : M \to \tilde M$ be a diffeomorphism.

Let $T \tilde M$ and $TM$ be tangent bundles of $\tilde M$ and $M$ respectively.

Let $\tilde \nabla$ be the connection in $T \tilde M$.

Let $\phi^* \tilde \nabla$ be the pullback connection.

Then $\phi^* \tilde \nabla$ is the connection in $TM$.