A similar question on MSE without answer.
Let $M, N$ be smooth manifolds such that $\partial N=\varnothing$. Let $A$ be a smoothly embedded submanifold of $N$ such that $\partial A\neq \varnothing$. Suppose, $f\colon M\to N$ be a smooth map with $f\pitchfork A,\ f\pitchfork \partial A$. In case $\partial M\neq \varnothing$, we also assume that the restriction $f\big|\partial M\to N$ is transverse to $A$.
Is it true that $f^{-1}(A)$ is a smoothly embedded submanifold of $M$ with $$\partial f^{-1}(A)=\begin{cases}f^{-1}(\partial A)&\text{ if }\partial M=\varnothing,\\ \big(f^{-1}(A)\cap \partial M\big)\cup f^{-1}(\partial A) &\text{ if }\partial M\neq \varnothing.\end{cases}$$
