Consider an $\left(\infty, 1\right)$-category $\mathcal{C}$ with a terminal object $1$. (I'm particularly interested in the case where $\mathcal{C}$ is a topos.) It is known that the forgetful functor $U$ from the under-category ${1}/{\mathcal{C}}$ to $\mathcal{C}$ creates colimits of diagrams with weakly contractible index categories. Is there an example of a diagram $F : \mathcal{I} \to {1}/{\mathcal{C}}$ such that


* $\mathcal{I}$ is *not* weakly contractible
* $U$ still creates the colimit of $F$?



Does the forgetful functor from a pointed $\left(\infty, 1\right)$-category only create weakly contractible colimits?
A short answer is that if $C$ is the terminal category, then $1/C \to C$ is an equivalence, and hence creates all colimits.


Less flippantly, we can take $C$ to be the ordinary topos of sets. Then the forgetful functor creates $I$-shaped colimits if $I$ is connected.


If $C$ is cocomplete, then the colimit of $F$ in $1/C$ is the pushout of a diagram
$$
1 \leftarrow colim\_I 1 \to colim\_I F
$$
and so the critical question is whether the colimit of the constant diagram with value $1$ is equivalent to $1$.


