If a composition of a surjective smooth function F with another function G between smooth manifolds is smooth, must G be smooth?

Question

It's easy to see that the composition of smooth functions between smooth manifolds is smooth. Let $M, N$, and $P$ be smooth manifolds of dimension $m$ $n$, $p$, respectively. If $F: M \rightarrow N$ is smooth and surjective, and $G \circ F: M \rightarrow P $ is smooth for some function $G: N \rightarrow P$, must $G$ be smooth? I suspect the answer is yes by analogy with the real case (we can just take derivative matrices, I think) but attempting to prove this becomes quickly fraught with domain issues ( I am still new to smooth manifold theory, so I could very well be wrong here) . Any help would be greatly appreciated!