@AdrianClough Note that $\pi^*A$ is equivalent to the colimit of $B\colon A\to \mathcal{X}$, where this is the unique functor factoring as $A\to \{*\} \to \mathcal{X}$. Thus you can define a functor $Fun(A,\mathcal{X})\to \mathcal{X}_{/\mathrm{colim} B}$ by sending $P\colon A\to \mathcal{X}$ to $(\mathrm{colim}P \to \mathrm{colim} B)$. The construction of this functor is natural in the infinity groupoid $A$, and it is clearly an equivalence if $A=*$. ...

Decent implies both $A\mapsto Fun(A,\mathcal{X})$ and $A\mapsto \mathcal{X}_{/\pi^*A}$ are limit preserving, so it follows that the functor is an equivalence for all infinity groupoids $A$.

@JonathanBeardsley Simplicial sets are a topos, so epimorphisms, extremal epimorphisms, regular epimorphisms, and quotients by congruence relations coincide. Every object $X$ has a small set of congruence relations (after all, $X \times X$ has a small set of subobjects) so $X$ has a small set of (extremal) epimorphic quotients. Alternatively, there's a theorem in Adamek-Rosicky that every locally presentable category is cowellpowered.

Is the category of coalgebras for a $\kappa$-accesible comonad on a $\kappa$-presentable category $\kappa$-presentable?

(same with $\infty$ sprinkled in)