Proof

For our purposes, a regular category is one that has finite limits and pullback-stable images.

The If ‘only if’ part of the proposition is straightforward: the adjunctions${\exists }_{f}C⊣f^{*}$ \exists_f C \dashv f^* come is from a pullbacks regular and category, images then in the adjunctions$C{\exists }_f⊣f^{*}$ C \exists_f \dashv f^* ( come from pullbacks and images in$C$ (Elephant lemma 1.3.1) as does the Frobenius property (${}$op. cit. lemma 1.3.3). The terminal object of $Sub(A) = Mon(C)_A$ is the identity $1_A$ on $A$, and binary products in the fibres $Sub(A)$ are given by pullback. The products are preserved by reindexing functors $f^*$ because (the $f^*$ are right adjoints but also because) a cone over the diagram for $f^*(S \wedge S')$ can be rearranged into a cone over that for $f^*S \wedge f^*S'$, giving the two the same universal property. The projection $cod$ clearly preserves these products. The Beck–Chevalley condition follows from pullback-stability of images in $C$.

… Conversely, suppose$Mon(C) \to C$ is a regular fibration. We need to show that $C$ has equalizers (to get finite limits) and pullback-stable images. But the equalizer of $f, g \colon A \rightrightarrows B$ is $(f,g)^*\Delta$. For images, let $im f = \exists_f 1$ as in Elephant lemma 1.3.1. Pullback-stability follows from the Beck–Chevalley condition.