The existence conjecture for direct images of higher-categorical families

Let $F:A\rightarrow B$ be a morphism of $n+1$-categories, and let $\varphi:A\rightarrow nCAT'$ be a family of $n$-categories over $A$. A direct image is a universal family $\psi:B\rightarrow nCAT'$ equipped with a morphism $\varphi\rightarrow F^*(\psi)$, denoted $F_*(\varphi)$. Direct-image conjecture. The direct image $F_*(\varphi)$ always exists and is essentially unique. The paper indicates that its construction should follow the earlier inverse-limit argument, subject to analogous verifications, but does not prove those verifications.