Pullbacks are indeed easier to define if you view a sheaf as a local homeomorphism. On the other hand, pushforwards are easier to define if you view a sheaf as a set-valued functor.
Tom Leinster
(Earlier up in the question there are a couple analogies to linear algebra. Matrices are computable whereas linear transformations are basis-free. Both define the same thing so it’s worth being able to think in either of the equivalent viewpoints.)













