In Chapter 7, Lemma 3, Milnor writes:
Lemma. If $f$ and $g$ are smoothly homotopic and $y$ is a regular value for both, then $f^{-1}(y)$ is framed cobordant to $g^{-1}(y)$.
Proof: Choose a homotopy $F$ with $$F(x, t)=f(x), 0\leq t<\epsilon\\F(x, t)=g(x), 1-\epsilon<t\leq 1.$$ Choose a regular value $z$ for $F$ which is close enough to $y$ so that $f^{-1}(z)$ is framed cobordant to $f^{-1}(y)$ and so that $g^{-1}(z)$ is framed cobordant to $g^{-1}(y)$. Then $F^{-1}(z)$ is a framed manifold and provides a framed cobordism between $f^{-1}(z)$ and $g^{-1}(z)$. This proves the lemma.
I wonder why he didn't say directly that $F^{-1}(y)$ is a framed manifold and provides a framed cobordism between $f^{-1}(y)$ and $g^{-1}(y)$? Is it right to say this?