if there exists a smooth map $M\rightarrow N$ and a smooth map $N\rightarrow M$ then $M,N$ are diffeomorphic

Question

Let $M,N\subseteq\mathbb R^n$ be smooth manifolds. Suppose there exists smooth maps $f:\ M\longrightarrow N$ and $g:\ N\longrightarrow M$ such that $f,g$ are bijective.

Here we only have the smoothness of the forward direction for each map. My query is if $M,N$ are diffeomorphic, which means there exists a bijection $h:\ M\longrightarrow N$ that’s smooth in both directions? If it’s not, is there a counterexample?