Family of Morse functions made constant I'm looking for a proof of the following theorem:
Let $f_t$ be a family of real-valued Morse functions defined on a smooth compact manifold $M$, and where $t$ is in $[0,1]$ (So for all value of $t$, $f_t$ is Morse). Also, $f_t$ depends smoothly on $t$.  Then, it is possible to find $\phi_t$ a family of diffeomorphism on $M$, and $\psi_t$ a family of diffeomorphism on $\mathbb{R}$, with $t$ in $[0,1]$ such that, $\phi_t$ and $\psi_t$ depends smoothly on $t$, and $\forall t \in [0,1] \psi_t \circ f_t \circ \phi_t = f_0$.
Does anyone know where to find a proof of this result?
Thank you for your help! 
 A: This is not an answer, but it will be too long for a comment.  It is just meant to be a intuition for why this is could (should?) be true.  
First, notice that we can rewrite your statement to $f_t = \psi_t^{-1}\circ f_0 \circ \phi_t ^{-1}$. Now, what you started this question with is a family of functions $\{f_t \}_{t \in I}$ that varies smoothly with $t$. I want to make the unproven claim that smoothly varying $M$ and smoothly varying $\mathbb{R}$ is exactly what $f_t = \psi_t^{-1}\circ f_0 \circ \phi_t ^{-1}$ is doing when we move through $t\in I$ for $f_t$. 
A slightly different approach, but with the same flavor, is to embed $M$ into $\mathbb{R}^n$, lets call that map $e$, and let $p$ be the projection onto the last coordinate. We can think of the Morse function, $f_t$ in the following way: 
$$ M\overset{\left(e,f_{t}\right)}{\longrightarrow}\mathbb{R}^{n}\times\mathbb{R}\overset{p}{\longrightarrow}\mathbb{R}
  $$
Then for each $t$ we have an embedding into Euclidean space such that the Morse function is projection onto the $(n+1)$-st axis. We know that $(e,f_t)$ is an smooth embedding since $M$ is compact and Euclidean space is Hausdorff. Then we can visualize the homotopy of our Morse functions as homotopy of our embedded $M$ with projection. The problem is this: do we know that homotopy of Morse functions preserves the level sets? I.e. Can we somehow have two critical points move past each other or possibly introduce a new critical point that will split a level set?  I am not sure, but I think there is a proof in here somewhere. Maybe someone who knows their differential topology better than I can give a full answer.
A: I  actually just found in the past days a proof given on the blog of a maths prof! It's a bit long to put it all here, plus the prof did all the job. Here's a link to the proof:
https://euclidlab.org/research/david-gays-blog?start=4
Thanks all for the help!
