My question concerns the *Propsition 5.12 of the paper of Bhatt-Mathew on arc-topology* where they claim that the functor $X\mapsto D^b_{\text{cons}}(X,\Lambda)^{[-n,n]}$ which assigns to a qcqs scheme $X$ the subcategory of the full derived category $D(X_{ét},\Lambda)$ of étale sheaves of $\Lambda$-modules of amplitude in $[-n,n]$ spanned by those objects which are bounded with constructible cohomology. Here they claim that this functor **satisfies $v$-descent and this automatically implies $v$-hyperdescent**. In the proof they use the fact this functor is bounded(in the derived category) and hence takes values in $\text{Cat}_{2n+2}$ the ($\infty$)-category of $2n+2$-categories. It seems to me that we can generalise the situation as follows.

Let functor $\mathcal{F}\colon\text{Sch}_R\to\mathcal{C}$(+ some properties) be a functor where $\mathcal{C}$ is an infinity category(+ some properties) and $\tau$ be a Grothendieck topology on $\text{Sch}_R$. Then I want to study the implication ``

$\tau$-descent $\implies$ $\tau$-hyperdescent''.

- Here I assume that $\mathcal{F}$ is locally of finite presentation(``finitary'' according to loc. cit.).
- In what generality of $\mathcal{C}$ is it is true ? I think I can prove the implication in small cases like when $\mathcal{C}=\text{Cat}_1$.

### Definitions

We say that $\mathcal{F}$ satisfies *$\tau$-descent* if for each $\tau$-covering $X'\to X$ the natural map $$
\mathcal{F}(X) \to\lim(\mathcal{F}(X')\rightrightarrows\mathcal{F}(X'\times_X X')\rightrightarrows \cdots)$$
is an equivalence in $\mathcal{C}$.
The functor is said to satisfy *$\tau$-hyperdescent* if for every hypercover $$X_{.}=(\cdots \rightrightarrows X_1 \rightrightarrows X_0 \to X_{-1}=X)$$ of $X$ in the $\tau$-topology we have that $$
\mathcal{F}(X)\to \lim(\mathcal{F}(X_0)\rightrightarrows \mathcal{F}(X_1) \rightrightarrows\cdots)$$ is an equivalence in $\mathcal{C}$.