I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist. 


**Remarks:**

1) A pullback in a certain category is defined as a space satisfying a universal property, not as a fiber product. (Which is just the usual form of many pullbacks...). See a clarification at the bottom.

2) I know [there are examples][1] where the pullback in the smooth category exists, but is different from the fiber product $Y \times_X Y' = \lbrace (y,y') \in Y \times Y'\mid f(y)=f'(y') \rbrace$ (which is always the pullback in the topological category). 

This is **not** what I am looking for, since in this example the smooth pullback (as the space satisfying the required universal property) exists.

3) In any case where the fiber product is a (smooth) submanifold, it is the pullback. Therefore, any possible example must be one whose fiber product is not a (smooth) submanifold. (In particular $f,f'$ [can't be transverse to each other][2]) 
   
4) [In this answer][3] there is a possible way of poving some limits does not exist. Maybe it's possible to use this method here also, but so far I didn't find an example

_______

**Clarification of the definition of pullback:**

 A space $Z$ (more precisely a diagram $Y \overset{g}{\leftarrow}Z\overset{g'}{\rightarrow}Y'$ which complete $Y\overset{f}{\rightarrow}X\overset{f'}{\leftarrow}Y'$ into a commutative square) is said to be a pullback if for any diagram $Y \overset{h}{\leftarrow}W\overset{h'}{\rightarrow}Y'$ (which commutes with $Y\overset{f}{\rightarrow}X\overset{f'}{\leftarrow}X$), there is a unique smooth/continuous map $u:W \to Z$ such that $h=g \circ u$ and $h'=g' \circ u$. (see [Wikipedia][4]).







[1]:http://mathoverflow.net/questions/124311/pullbacks-as-manifolds-versus-ones-as-topological-spaces
[2]:http://mathoverflow.net/a/1067/46290
[3]:http://mathoverflow.net/a/19473/46290
[4]:https://en.wikipedia.org/wiki/Pullback_%28category_theory%29