The universal morphism conjecture for properties of higher-categorical morphisms

Let $\psi:A\rightarrow nCAT'$ be a functor and let $\mathcal P$ be a property of morphisms from $\psi$ to objects of $nCAT'$. A universal morphism with property $\mathcal P$ is a morphism $\nu:\psi\to U$ that is universal among morphisms having that property. Universal morphism conjecture. If the four conditions listed in the paper hold—preservation of $\mathcal P$ by inverse limits, bounded factorization, preservation of $\mathcal P$ by the telescope construction, and the stated mapping criterion—then the argument of Section 5 constructs a universal morphism $\nu:\psi\to T$ with property $\mathcal P$. This is a conditional construction rather than an unconditional existence claim, and the paper does not establish all four conditions.