Consider a pair of morphisms $A \to_f C \leftarrow_g B$. The pull-back of f along g is a triple A$A \leftarrow_p D \to_q B$ such that $f \circ p = g \circ q$ and, for any $A \leftarrow_{p'} D' \to_{q'} B$ such that $f \circ p' = g \circ q'$, there exists a unique $h : D' \to D$ such that $p' = p \circ h$ and $q' = q \circ h$. Diagrammatically,

Often the object $D$ itself is called the pullback

In the category Set, it corresponds to the subset of the Cartesian product:

$A \times_C B = \{(a,b) \in A \times B | f(a) = g(b) \}$

so it semantically corresponds to an Equation

Existence of arrow $h$ means that $D$ must have enough capacity (enough elements) to transmit the information. So it should have at least the cardinality of the set above (because the functions p' and q' which satisfy the commuting diagram, can only take elements in D' to pairs that satisfy the condition $f(a)=f(b)$ for it to commute).

Uniqueness of $h$ means that $D$ doesn't have too much capacity. So that there is not arbitrary choices because different elements of $D$ "do the same thing" (by being mapped to same elements by p and q). Therefore, the set $D$ has the same cardinality as the set $A \times_C B$ and so it's the same up to unique Isomorphism (bijection).

Furthermore p and q should act as projections.

https://ncatlab.org/nlab/show/pullback