1. Is there a relationship between the pullback in differential geometry and the pullback in category theory?
[2. Is there a relationship between the pushforward/pushout in differential geometry and the pushforward/pushout in category theory? Although the answer to the above (1.) is equivalent to the answer to this question (2.) by duality.]
As far as I can tell, (for the terms in differential geometry) the pullback is a contravariant functor, and the pushforward is a covariant functor.
3. Is there a way to turn every contravariant functor into a category theory pullback or vice versa? (Or every covariant functor into a category theory pushforward or vice versa?)
If the answer to 3. is no, then it seems like the fact that the two concepts have the same name is just a historical accident, and does not indicate that one is meant to generalize the other.
4. Is it true that the similar terminology between the two fields is a historical accident? Or are they both meant to evoke the same type of basic example?