I am looking for a good set of sufficient conditions to put on a category $C$ such that the following becomes true:


Given any $\infty$-category $S$ and a functor $f\colon N(C)\to S$ which is essentially constant (i.e. it maps all edges to equivalences), a colimit $\overline f\colon N(C)^\triangleright\to S$ of $f$ exists and is always an essentially constant diagram.



Conditions on a diagram category such that constant limits always exist and is constant
I think that Proposition 4.3.1.12. in Higher Topos Theory (Lurie) is exactly what I was looking for:


Given: a coCartesian fibration $p\colon X\to S$ of $\infty$-categories and a diagram $\overline q\colon K^\triangleright\to X$ such that $K$ is weakly contractible and such that $\overline q$ carries each edge of $K$ to a $p$-coCartesian edge of $X$.
Then: $\overline q$ is a $p$-colimit diagram if and only if it carries all edges in $K^\triangleright$ to $p$-coCartesian edges of $X$.


Taking $S$ to be a point we get that a sufficient condition as I was looking is for $N(C)$ to be weakly contractible.


It seems that Kyle Ferendo's remark implies the converse: if all the essentially constant diagrams of a certain shape are essentially constant, then $N(C)$ is weakly contractible. (Because $|N(C)|$ is the limit of the constant diagram $\star$ in spaces)


