I'm reading actually wiki's article about the Descent construction for vector bundles.
After motivating the idea as a generalization of "gluing" vector bundles over topological spaces there is introduced a more abstract approach:
Since the disjoint union $Y= \cup X_i $ one introduce the disjoint union of $X_{ij} = X_i \cap X_j$ as fiber product
$$Y \times_X Y$$
That's also ok since considering what happens on the intuitive intersection $X_{ij} = X_i \cap X_j$ is in light of category theory comparing the canonical projections $p_i: Y = \cup X_k \to X_i \subset X$
using equalizer of two copies of the projection $p$. The bundles on the $X_{ij}$ that have to be controlled are $V'$ and $V''$.
Now my question: In the article $V'$ and $V''$ are called the "pullbacks" to the fiber of $V$ via the two different projection maps to $X$.
I don't understand this interpretation of the $V'$ and $V''$. How they here are interpreted as pullbacks in categorical sense? Can anybody show the corresponding cartesian diagram for them? Furthermore, what is here "the fiber of $V$"? Especially what does the expression "pullback to the fiber of blabla" mean in this context?
Here the full article: https://en.wikipedia.org/wiki/Descent_(mathematics)