(Sketch) This is a special case of the usual proof that Conduché functors are the exponentiables in \(\mathbf {Cat}\), as in e.g. Street's Powerful functors, although some details of exponentiable implies Conduché are omitted there and sketched below.
The key point is that when \(p : E \to B\) has discrete fibres, the profunctors \(m_E(\beta ) : E_b' \to E_b\) recording morphisms over \(\beta \) for each \(\beta : b \to b'\), amount to spans of sets, and asking for \(m_E(\beta ');m_E(\beta ) \to m_E(\beta ';\beta )\) to be an isomorphism of spans is exactly the discrete Conduché (unique lifting of factorizations) condition.
When this map is an isomorphism, the underlying data of the putative exponential is indeed a functor into \(B\) from a well-defined category, as shown in Street, establishing Conduché \(\Rightarrow \) exponentiable. A proof along the same lines with less machinery is in Johnstone, Fibrations and partial products in a 2-category.
Conversely, let the existence of a right adjoint \((-)^p : \mathbf {Cat} \to \mathbf {Cat} / B\) to pullback be given (the sufficiency of this reduction is shown in Street), and hence \(X^p : Z \to B\) is a functor for every category \(X\). From the fact that \(Z\) is a category, we can derive that \(p\) is discrete Conduché as follows.
Let \(f : x \to z\) be a morphism of \(E\) whose image factorizes as \(p(f) = \gamma \circ \delta \), with \(\delta : p(x) \to y, \gamma : y \to p(z)\).
For the functors \(b : 1 \to B\), \(\gamma ,\delta : 2 \to B\), consider the diagram \(D\) in \(\mathbf {Cat} / B\),
whose pushout is \((\gamma ,\delta ) : 3 \to B\), picking out \(\gamma ,\delta \) and the composite \(\gamma \circ \delta \).
Since by assumption \(p^{*}\) is a left adjoint, it preserves this pushout. The colimit of the diagram \(p^{*} \circ D\) is the category containing x,z and the fibre over y, with morphisms between fibres given by all maps over \(\gamma \) and over \(\delta \), and closed under composition. On the other hand \(p^{*}\) sends the colimit of \(D\) to a category which additionally contains all maps over \(\gamma \circ \delta \), and hence preservation of the colimit says that every map over \(\gamma \circ \delta \) is a composite of a map over \(\gamma \) and a map over \(\delta \). This establishes lifting of factorizations.
Uniqueness follows from the assumed discreteness of the fibres of \(p\).