Let $f:Y\rightarrow W,g:Z\rightarrow W$ are submersions (so that $Y\times_W Z$ is a manifold).
Suppose $F:X\rightarrow Y\times_WZ$ is a smooth map such that $pr_1\circ F:X\rightarrow Y$ is submersion and $pr_2\circ F:X\rightarrow Z$ is a submersion. Then, I want to prove that $F$ is a submersion.
I think this has to be true. Can some one give a quick proof.
Here $Y\times_WZ$ is the pullback manifold $Y\times_WZ=\{(y,z):f(y)=g(z)\}\subseteq Y\times Z$.
I am sorry but it is not true. Let us consider the following counterexample. Let $W=\{*\}$ be the singleton, let $Y=Z=X$ be a manifold of dimension greater or equal to $1$ and let $f=g$ be the unique possible map. Then the pullback $Y\times_W Z$ is simply the usual product $X\times X$. Finally, let $F=\Delta : x \in X \mapsto (x,x)\in X\times X$ be the diagonal map, which is not a submersion. However, $pr_1 \circ F = pr_2 \circ F = id_X$ is a submersion.