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For a site $\mathcal{C}$, an $n$-cosheaf is a functor $\mathcal{F}:\mathcal{C}\rightarrow $n$\text{-}\mathfrak{Cat}$ such that for any cover $(U_i\rightarrow U)_i$, we have $$\prod_{i,j}\mathcal{F}(U_i\times_U U_j)\rightrightarrows \prod_i \mathcal{F}(U_i)\rightarrow \mathcal{F}(U)$$ is an $n$-pushout. Does it then follow that the contravariant functor $$H_{\mathcal{F}}:\mathcal{C}^{op}\rightarrow n\text{-}\mathfrak{Cat},\quad X\mapsto \text{Hom}_{n\text{-}\mathfrak{Cat}}(\mathcal{F}(X),\mathfrak{C})$$ is a $n$-sheaf, where $\mathfrak{C}$ is a $n$-category? Basically what I'm asking is if the "$n$-Hom" functor sends pushouts to pullbacks.

The case I'm most interested in is that of cosheaves of $\infty$-groupoids and groupoids.

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    $\begingroup$ Yes, Homs send colimits to limits always. $\endgroup$
    – Qiaochu Yuan
    Commented Sep 21, 2020 at 19:16
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    $\begingroup$ I agree with Qiaochu, modulo the fact that your definition of an $n$-cosheaf is wrong. The right definition of a sheaf valued in an $\infty$-category $C$ says that for any cover $U_\bullet \to U$, we have that $\mathcal F(U)$ is canonically equivalent to the $\infty$-categorical limit of the functor $\Delta \to C$, $[n] \mapsto \prod_{i_0,\dots,i_n} \mathcal F(U_{i_0} \times_U \cdots \times_U U_{i_n})$. If $C$ is an $n+1$-category like $Cat_n$, then I believe you can truncate this simplicial object at level $n+1$ and still get things right. Cosheaves should be defined by the analogous colimit. $\endgroup$
    – Tim Campion
    Commented Sep 21, 2020 at 22:08

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