First note that \mathcal {C}^{[n]} \to \mathcal {C}^{\partial \Delta ^n} = M_n\mathcal {C} is an isomorphism when n \geq 4, since a pseudofunctor involves only 3-dimensional information. In this case the functor \mathcal {C}^{[n]} \to M_n\mathcal {C} \times _{M_n\mathcal {D}} \mathcal {D} is an isomorphism, and so certainly a fibration.

When n=3, the objects of the codomain correspond to triples of morphisms f,g,h with two choices of composite, and such an object is in the image if and only if the given diagram of 2-cells commutes---the functor is locally an isomorphism. Hence the functor \mathcal {C}^{[3]} \to M_3\mathcal {C} \times _{M_3\mathcal {D}} \mathcal {D} is locally an isomorphism, so it suffices to prove it has the equivalence lifting property for 1-cells.

To see this, observe that since \mathcal {D}^{[3]} \to M_3\mathcal {D} is a local isomorphism, to give a morphism in the pullback is just to give a morphism in M_3\mathcal {C}. But since the property of commuting is certainly stable under equivalence, indeed we can lift such a morphism.

When n=2, the objects of the matching bicategory M_2\mathcal {C} are simply triangles, and to lift a triangle is to choose a 2-isomorphism rendering it commutative. Now \mathcal {C}^{[2]} \to M_2\mathcal {C} merely has discrete fibers.

In this case, let \sigma : [2] \to \mathcal {C} be a pseudofunctor, i.e a commutative triangle. An isomorphism starting at the image of \sigma in the pullback M_2\mathcal {C} \times _{M_2\mathcal {D}}\mathcal {D}^{[2]} consists of

1. A choice of three 2-isomorphisms in \mathcal {C}, one for each side
2. A choice of isomorphism p(\sigma ) \to \gamma ' in \mathcal {D}^{[2]} (consisting of a 2-isomorphism for each side, so that composing them all with the 2-isomorphism filling p(\sigma ) yields the one filling p(\gamma ))
3. Such that the 2-isomorphisms in \mathcal {C} are taken to the ones in \mathcal {D}

There is only one possible lift, namely the isomorphism \sigma \to \gamma where \gamma is defined as the composite of the given 2-isomorphisms. But now commutativity of the square precisely implies this must be a lift of the isomorphism in \mathcal {D}^{[2]} as well, and we're done.

Finally, the case of n=1 is completely analogous to n=2. M_1\mathcal {C} \cong \mathcal {C}^2, and the fiber over two objects is the hom-category. Here we use the isomorphism lifting property for the individual hom-categories.