Let $f:M_0\rightarrow M_1$ be a diffeomorphism between two compact $n$-dimensional manifolds. Let $W^{n+1}$ be an $h$-cobordism between $M_0$ and $M_1$. Assume that the cobordism has no torsion and its dimension is high enough so that, by the $s$-cobordism theorem, there is a diffeomorphism $F:M_0\times I\rightarrow W$. Can $F$ be taken so that $F|_{M_0\times 0}=id_{M_0}$ and $F|_{M_0\times1}=f$?

Thanks in advance!