objects in an arbitrary $\infty$-topos ${\mathcal}{X}$. Our first result (cf. Theorem \[thm:CSSaSheaf\]) in this paper is to show that the construction $$\label{eq:CSSfunctor}{\mathcal}{X}\mapsto {CSS}_k({\mathcal}{X})$$ satisfies a certain descent condition: suppose that ${\mathcal}{X}_i\to {\mathcal}{X}$ is an étale cover of ${\mathcal}{X}$ indexed by a small simplicial set $i\in I$, i.e. $${\mathcal}{X}\cong {{{\operatorname}{co}\lim}}_{i\in I}{\mathcal}{X}_i,$$ then $${CSS}_k({\mathcal}{X})\cong \lim_{i\in I}{CSS}_k({\mathcal}{X}_i).$$ In other words, defines a stack (cf. [@Lurie:2009un Notation 6.3.5.19]). Given an $\infty$-category ${\mathcal}{C}$, Barwick [@Barwick:2013th] showed how to construct an $(\infty,1)$-category ${\operatorname}{Span}({\mathcal}{C})$ which has the same space of objects as ${\mathcal}{C}$, but whose morphisms between two objects $c_0,c_1\in {\mathcal}{C}$ is the space of diagrams in ${\mathcal}{C}$ of the form $$\begin{tikzpicture}
 \node  (m-1-1)  at (0,1) {$x$};
 \node  (m-1-2)  at (1,0) {$c_0$};
 \node  (m-2-1)  at (-1,0) {$c_1$};
\draw[->] (m-1-1) -- (m-1-2);
\draw[->] (m-1-1) -- (m-2-1);
\end{tikzpicture}$$ That is, spans $c_0\nrightarrow c_1$ in ${\mathcal}{C}$. Composition of two such morphisms is given by taking the fibred product. Given an $\infty$-topos ${\mathcal}{X}$ and a complete $k$-fold Segal object $X_{\bullet,\dots,\bullet}\in{CSS}_k({\mathcal}{X})$, Haugseng extends this construction in [@Haugseng:2014vw] to produce an $(\infty,k)$-category ${\operatorname}{Span}_k({\mathcal}{X},X_{\bullet,\dots,\bullet})\in{CSS}_k(\hat{\mathscr{S}})$ of iterated $k$-fold spans with local systems valued in $X_{\bullet,\dots,\bullet}\in{CSS}_k({\mathcal}{X})$ (here $\hat{\mathscr{S}}$ is the $\infty$-category of (not necessary small) spaces).

Our second main result is to show that the assignment 