Let $\mathsf{A},\mathsf{B},$ and $\mathsf{C}$ be (ordinary) categories and $F : \mathsf{A}\to\mathsf{C}$ and $G : \mathsf{B}\to\mathsf{C}$ be functors such that
- $\mathsf{A}$ and $\mathsf{B}$ are complete, and
- $G$ preserves limits.
Then the comma category $(F\downarrow G)$ is complete, and both forgetful functors $U : (F\downarrow G)\to\mathsf{A}$ and $V : (F\downarrow G)\to\mathsf{B}$ preserve limits.
To prove that $(F\downarrow G)$ is complete, we first use the fact that given any diagram $p : K\to(F\downarrow G),$ the compositions $U\circ p$ and $V\circ p$ admit limits in $\mathsf{A}$ and $\mathsf{B},$ and then observe that in fact we can produce a cone on $G\circ V\circ p$ by "applying $F$ to the limit of $U\circ p.$" We can prove this statement by hand, using the data of the compatible morphisms from the limit object of $U\circ p$ to the $U\circ p(k)$ for $k\in K.$ Then, since $G$ preserves limits, the morphism of cones from the cone just constructed to the limit cone on $G\circ V\circ p$ gives us an element of $(F\downarrow G),$ which we then prove is actually the limit of $p.$
If $\mathsf{A},\mathsf{B},$ and $\mathsf{C}$ are replaced by $\infty$-categories $\mathcal{A},\mathcal{B}$ and $\mathcal{C},$ and we consider the comma $\infty$-category $(F\downarrow G)$ which is the pullback of $\mathcal{A}\times\mathcal{B}\xrightarrow{F\times G}\mathcal{C}\times\mathcal{C}\xleftarrow{(s,t)}\mathcal{C}^{\Delta^1},$ I don't see how to make the same argument work. In particular, the step where I need to apply $F$ to the limit of $U\circ p$ to produce a cone on $G\circ V\circ p$ is unclear.
If I begin with a map $p : K\to (F\downarrow G)$ where $K$ is a simplicial set, I interpret the limits of $U\circ p$ and $V\circ p$ as extensions $p_U : K^\triangleleft\to\mathcal{A}$ and $p_V : K^\triangleleft\to\mathcal{B}.$ However, I can't seem to make sense of how to translate the step of the argument where I "apply $F$ to $p_U$" to produce a cone on $G\circ V\circ p.$
I would greatly appreciate any advice regarding how to translate the $1$-categorical argument above into the $\infty$-categorical setting (or a heads-up if what I'm trying to prove is no longer true and I need additional assumptions.)
Note: I'm actually more interested in the dual statement for cocompleteness and preservation of colimits, but the limit statement is interesting to me as well. I would also be happy to assume that all of $\mathcal{A},\mathcal{B},$ and $\mathcal{C}$ are presentable if that makes the proof easier.
