Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth.
If any of $f$ and $g$ is smooth can we conclude that the other is also smooth? In particular, is $f$ smooth if $g$ is a smooth surjection? Is any of the preceding statements a case of a categorical theorem? What conditions should $f$ and $g$ satisfy for the first statement to hold true? What conditions should $f$ and $g$ satisfy for the second statement to be true?
No.
Consider for surjective smooth $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ the projection onto the first factor ($(x,y)\rightarrow x$) and, for $x\in \mathbb{R}$, $g(x) = (x,1)$ if $x<0$ and $= (x, -1)$ if $x\ge 0$. Then $g$ is not even continuous, but $f \circ g(x) = x$.
(Edit: you can easily modify $g$ from the above example in an obvious fashion so that it is not even continous at any given point and all other statements remain valid).