TITLE: Colimits glue. What do limits do?
QUESTION [5 upvotes]: The informal but very useful way to think of colimits is as 'gluing things'. This intuitive view is very helpful to me, and I'd like one for limits aswell, but I haven't stumbled opon such a thing anywhere so...

If colimits glue, what do limits do?

REPLY [2 votes]: Sheaves provide a nice framework for getting comfortable with the notion of a limit. See my answer at the relevant MO discussion
MO/23268: Geometric intuition for limits
There you will also find many other interesting answers.
To add something "new" and explain a little bit the other answers: Compute the pullback of the two obvious maps $(S^2)^+ \to S^2 \leftarrow (S^2)^-$, where by $(S^2)^+$ I mean the upper hemisphere and by $(S^2)^-$ the lower hemisphere.