How can the subobject fibration be obtained from the codomain fibration Suppose $\mathcal A$ is category with pullbacks. Consider the codomain fibration $F : \mathcal A^\to \to \mathcal A$ and the restriction $F' : \mathcal{A}^\to_{\operatorname{Mon}} \to \mathcal A$ on the full subcategory of monos of $\mathcal A$.

Is there are neat description / construction of $F'$ in terms of $F$ of some 'fibrational' construction that 'immediately' shows that $F'$ is a fibration too?

(in the 'immediately' part we can assume something like "monos are stable under pullbacks")

Motivation: One may want to replace "all monos" with "certain monos" and look, whether we still get a fibration.
 A: One characterization is the following.
A monomorphism into an object $X$ inside a category $\mathcal C$ is (by definition) a subterminal object of the overcategory $\mathcal C/X$ (i.e. it admits at most one morphism from any other object in the fiber over $X$ of the codomain fibration). Since the codomain fibration is a bifibration i.e. since the pullback functors have left adjoints (post-composition functors), the pullback functors preserve limits and monomorphisms. In particular they preserve subterminal objects in the fibers, hence the subterminal objects in the fibers generate a full subfibration.
So one way to generalize to fibrations the statement that "monomorphisms are stable under pullback" is via the statement that the restriction of a fibration $\mathcal C\xleftarrow{p}\mathcal F$ to the full subcategory $\mathcal M\hookrightarrow\mathcal F$ whose objects are the subterminal objects in fibers of $\mathcal C\xleftarrow{p}\mathcal F$ is itself a fibration. In particular, this is the case if $\mathcal C\xleftarrow{p}\mathcal F$ is a bifibration, and more generally if the pullback functors of $\mathcal C\xleftarrow{p}\mathcal F$ preserve subterminal objects.
