Let $M$ be a smooth manifold (without boundary). Suppose $F:M \times I \rightarrow M$ is a smooth (say $C^{\infty}$) isotopy between two diffeomorphisms $f_0$ and $f_1$ of M in the sense that $\forall t \in I ,F(\cdot,t):M\rightarrow M $ is an embedding and that $F(\cdot,0)=f_0,F(\cdot,1)=f_1$. My question is can we find a new isotopy $\tilde{F}$ between $f_0$ and $f_1$ such that $\forall t \in I,\tilde{F}(\cdot,t) $ is a diffeomorphism?
For exmaple if M is a compact manifold. By the isotopy extension theorem, there is an ambient isotopy $\Phi:M\times I \rightarrow M$ such that $\Phi(\cdot,0)=id_M,\Phi(1,f_0(\cdot))=f_1(\cdot)$. Then $\Phi(\cdot,f_0(\cdot))$ is an isotopy between $f_0$ and $f_1$ through diffeomorphism. I wonder if this is true in general. Thank you!