Let consider a complete regular category $\mathscr C$. For $x:X\to A$ and $y:Y\to A$ we write $x\equiv y$ if there exists regular epimorphisms $u:W\to X$ and $v:W\to Y$ such that $x\circ u=y\circ v$.
Let $\Phi$ be a diagram in $\scr C$ indexed by a small category $\scr I$. Assume that for each object $n$ of $\scr I$ is given a morphism $\phi_n:D_n\to\Phi_n$ such that for each $i:n\to m$ in $\scr I$ we have $\phi_m\equiv\Phi(i)\circ \phi_n$.
Let $\lambda_n:\lim\Phi\to\Phi_n$ be a limit cone.
Question: there exists a morphism $\phi:D\to\lim\Phi$ such that $\phi_n\equiv\lambda_n\circ\phi$?
Clearly the answer is yes if we require $\phi_m=\Phi(i)\circ\phi_n$.