  In this article we present a solution to a conjecture of Vladimir Voevodsky
regarding C-systems. This conjecture provides, under some assumptions, a lift
of a functor $M\colon \mathrm{CC} \rightarrow \mathcal{C}$, where $\mathrm{CC}$
is a C-system and $\mathcal{C}$ a category, to a morphism of C-systems
$M'\colon \mathrm{CC} \rightarrow \mathrm{CC}(\widehat{\mathcal{C}},p_M)$. We
explain the motivation behind this conjecture and introduce the required
background material on C-systems. Finally, we give a proof of this conjecture.
