A Conduché functor, or exponentiable functor, is a functor which is an exponentiable morphism in Cat. (In accordance with Baez's law, the notion was actually defined by Giraud before Conduché.) This turns out to be equivalent to a certain “factorization lifting” property which includes both Grothendieck fibrations and opfibrations.
As is evident from the fact that such functors have a name, not every functor is exponentiable in $Cat$. In particular, although $Cat$ is cartesian closed, it is not locally cartesian closed.
It is easy to write down examples of colimits in $Cat$ that are not preserved by pullback (as they would be if pullback had a right adjoint). For instance, let $\mathbf{2}$ denote the walking arrow, i.e. the ordinal $2$ regarded as a category, $1$ the terminal category, and $\mathbf{3} = \mathbf{2} \sqcup_1 \mathbf{2}$ the ordinal $3 = (a \to b \to c)$ regarded as a category. Then the pushout square
in $Cat/\mathbf{3}$ pulls back along the inclusion $\mathbf{2}\to \mathbf{3}$ of the arrow $(a\to c)$ to the square
which is certainly not a pushout.
One way to describe the problem is that the pushout has “created new morphisms” that didn’t exist before. But another way to describe the problem is that the inclusion $\mathbf{2}\to\mathbf{3}$ fails to notice that the morphism $(a\to c)$ acquires a new factorization in $\mathbf{3}$ which it didn’t have in $\mathbf{2}$. Conduché’s observation was that this latter failure is really the only problem that can prevent a functor from being exponentiable.
A functor $p\colon E\to B$ is a strict Conduché functor if for any morphism $\alpha\colon a\to b$ in $E$ and any factorization $p a \overset{\beta}{\to} c \overset{\gamma}{\to} p b$ of $p \alpha$ in $B$, we have:
there exists a factorization $a \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b$ of $\alpha$ in $E$ such that $p \tilde{\beta} = \beta$ and $p \tilde{\gamma} = \gamma$, and
any two such factorizations in $E$ are connected by a zigzag of commuting morphisms which map to $id_c$ in $B$.
(Here, ‘commuting morphism’ means a morphism $d \to d'$ in $E$ such that the pair of triangles in
commute.)
The theorem is then that the following are equivalent:
By “exponentiable in the strict 2-category $Cat$” we mean that pullback along $p$ has a strict right 2-adjoint (i.e. a $Cat$-enriched right adjoint). Of course, this implies ordinary exponentiability in the 1-category $Cat$, while the converse follows via an argument involving cotensors with $\mathbf{2}$ in $Cat$.
For exponentiability in the weak 2-category $Cat$, in the sense of pullback having a weak/pseudo 2-adjoint, we can simply weaken the condition. We say that $p\colon E\to B$ is a (weak) Conduché functor if for any morphism $\alpha\colon a\to b$ in $E$ and any factorization $p a \overset{\beta}{\to} c \overset{\gamma}{\to} p b$ of $p \alpha$ in $B$, we have:
there exists a factorization $a \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b$ of $\alpha$ in $E$, and an isomorphism $p d \cong c$, such that modulo this isomorphism $p \tilde{\beta} = \beta$ and $p \tilde{\gamma} = \gamma$, and
any two such factorizations in $E$ are connected by a zigzag of commuting morphisms which map to isomorphisms in $B$.
A functor can then be shown to be a weak Conduché functor if and only if it is exponentiable in the weak sense in $Cat$.
The Conduché criterion can be reformulated in a more conceptual way by analogy with Grothendieck fibrations. We first observe that to give a functor $p\colon E\to B$ is essentially the same as to give a normal lax 2-functor $B\to Prof$ from $B$ to the 2-category of profunctors. Specifically, given a functor $p$, we define $B\to Prof$ as follows. Each object $b\in B$ is sent to the fiber category $p^{-1}(b)$ of objects lying over $b$ and morphism lying over $1_b$. And each morphism $f\colon a\to b$ in $B$ to the profunctor $H_f\colon p^{-1}(a) ⇸ p^{-1}(b)$ for which $H_f(x,y)$ is the set of arrows $x\to y$ in $E$ lying over $f$. The lax structure maps $H_f \otimes H_g \to H_{g f}$ are given by composition in $E$. The converse construction of a functor $p$ from a normal lax 2-functor into $Prof$ is an evident generalization of the Grothendieck construction. Now we can say that:
Non-strict Conduché functors and Street fibrations may be equivalently characterized by an “up-to-iso” version of the above construction using essential fibers.
The above considerations show that any Grothendieck fibration or opfibration is a (strict) Conduché functor, while any Street fibration or opfibration is a non-strict Conduché functor.
If $\mathbf{2}$ denotes the interval category, then any normal lax functor out of $\mathbf{2}$ is necessarily pseudo, since there are no composable pairs of nonidentity arrows in $\mathbf{2}$. It follows that, as pointed out by Jean Benabou, any functor with codomain $\mathbf{2}$ is a Conduché functor. Note that functors with codomain $\mathbf{2}$ can also be identified with profunctors, the two fiber categories being the source and target of the corresponding profunctor.
As with exponentiable morphisms in any category, Conduché functors are closed under composition.
F. Conduché, ‘Au sujet de l’existence d’adjoints à droite aux foncteurs “image réciproque” dans la catégorie des catégories’, C.R. Acad. Sci. Paris 275 (1972), A891–894.
J. Giraud, ‘Méthode de la descente’, Bull. Math. Soc. Memoire 2 (1964)
The definitions and proofs of the above theorems, along with the 2-categorical generalization (Conduché considered only the 1-categorical case) can also be found in
A description of the characterization in terms of lax normal functors can be found in