# Prop 5.3.6.2 in higher topos theory

Here are the necessary notations and the statement of the proposition. I don't understand why the underlined sentence is true. Are the horizontal morphisms Cartesian fibration?

I would appreciate it if someone could give me some hints.

The two vertical maps are fully faithful, because $$D\subset D'$$ is, and it preserves $$\mathcal K$$-indexed colimits.
Therefore, the pullback of the right vertical morphism is also a full subcategory $$E$$ of $$Fun_\mathcal K(P_\mathcal R^\mathcal K(C),D')$$, and by the commutativity of the diagram, $$Fun_\mathcal K(P_\mathcal R^\mathcal K(C),D)\subset E$$ as full subcategories.
The statement that the square is cartesian is equivalent to the statement that this inclusion be an equivalence, i.e. that any object of $$E$$ be in the image of $$Fun_\mathcal K(P_\mathcal R^\mathcal K(C),D)$$. But objects of $$E$$ are exactly objects of $$Fun_\mathcal K(P_\mathcal R^\mathcal K(C),D')$$ whose restriction along $$j$$ lands in $$D$$, so this is equivalent to what Lurie claims.