Surjection of a short exact sequence induced by spectral sequence (from paper of Schneider/Stuhler) Let $K=\mathbb{Q}_p$ and $X$ a smooth separated rigid analytic variety over $K$ with coherent sheaf $\mathcal{F}$. Furthermore, $U \subset X$ is an open subvariety with admissible covering
$$ \ldots \subset U_n \subset U_{n+1} \ldots $$
of open subvarieties. Let $Z:=X \backslash U$ and $Z_n:=X \backslash U_n$, i.e. we have
$$ \ldots \supset Z_n \supset Z_{n+1} \ldots $$ and  $\bigcap_{n \in \mathbb{N}}Z_n=Z.$
Then, Proposition 4 in Section 2 of https://ivv5hpp.uni-muenster.de/u/pschnei/publ/pap/xsymm.pdf
gives a short exact sequence
$$ 0 \rightarrow {\varprojlim_n}^{(1)}H^{*-1}_{Z_n}(X,\mathcal{F}) \rightarrow H^{*}_{Z}(X,\mathcal{F}) \stackrel{g}{\rightarrow} {\varprojlim_n}H^{*}_{Z_n}(X,\mathcal{F}) \rightarrow 0.$$
The proof of that Proposition tells us, that the sequence is induced by considering two standard spectral sequences for the hypercohomology of the functor $\varprojlim$.
I was wondering if the surjection $g:H^{*}_{Z}(X,\mathcal{F}) {\rightarrow} {\varprojlim_n}H^{*}_{Z_n}(X,\mathcal{F})$ is the morphism induced by the universal property of the inverse limit applied to the natural composition
$$H^{*}_{Z}(X,\mathcal{F}) \rightarrow H^{*}_{Z_{n+1}}(X,\mathcal{F}) 
\stackrel{f_{n,n+1}}{\rightarrow} H^{*}_{Z_{n}}(X,\mathcal{F})$$
coming from the chain $Z \subset Z_{n+1} \subset Z_{n}$. Here, the $f_{n,n+1}$ are the transition maps defining the inverse limit ${\varprojlim_n}H^{*}_{Z_n}(X,\mathcal{F}).$ Or equivalently, is
$g$ composed with the natural projection
$$p_n:{\varprojlim_n}H^{*}_{Z_n}(X,\mathcal{F}) \rightarrow H^{*}_{Z_{n}}(X,\mathcal{F})$$ the natural morphism $\varphi:H^{*}_{Z}(X,\mathcal{F}) \rightarrow H^{*}_{Z_{n}}(X,\mathcal{F})$ induced by the inclusion $Z \subset Z_n$?
 A: Let's see if this works.
So I think the answer to my question is positive.
Let's take an injective resolution $\mathcal{F}\rightarrow \mathcal{I}^\bullet$ of $\mathcal{F}$ an let $K_n^\bullet:=H^0_{Z_n}(X,\mathcal{I}^\bullet)$ be a complex which computes $H^*_{Z_n}(X,\mathcal{F})$. Then be $K_n^\bullet\rightarrow J_n^{\bullet,\bullet}$ a Cartan-Eilenberg resolution of $K_n^\bullet$, such that $K^p\rightarrow J^{p,\bullet}$ is an injective resolution. Then, by definition, $$H^*(\mathrm{Tot}(\varprojlim_{n}J_n^{\bullet,\bullet}))=H^*(R\varprojlim_nK_n^\bullet).$$
The double complex $\varprojlim_n J_n^{\bullet,\bullet}$ induces by the two standard filtrations the mentioned hypercohomology spectral sequences
\begin{align}
{}^IE_1^{pq}&={\varprojlim_n}^{(q)}H^0_{Z_n}(X,\mathcal{I}^p) \Rightarrow H^{p+q}(R\varprojlim_nK_n^\bullet), \\
{}^{II}E_2^{pq}&={\varprojlim_n}^{(p)}H^q\big(H^0_{Z_n}(X,\mathcal{I}^\bullet)\big) \Rightarrow H^{p+q}(R\varprojlim_nK_n^\bullet).
\end{align}
By the arguments from the Proof of Proposition 4 in Section 2 of https://ivv5hpp.uni-muenster.de/u/pschnei/publ/pap/xsymm.pdf, we know
$${}^IE_1^{pq}=0 \text{ for } q \neq 0, \,\,\,\,\,\,\,\,\,(1)$$
$$ {}^{II}E_2^{pq}=0 \text{ for } p \neq 0,1. \,\,\,(2)$$
Hence, both spectral sequences degenerate on the second page.
Additionally, let us fix $n \in \mathbb{N}$ and set
\begin{align}
{}^IE_0^{'pq}&=J_n^{p,q} \Rightarrow H^{p+q}\big(\mathrm{Tot}(J_n^{\bullet,\bullet})\big)=H^{p+q}_{Z_n}(X,\mathcal{F}), \\
{}^{II}E_0^{'pq}&=J_n^{q,p} \Rightarrow H^{p+q}\big(\mathrm{Tot}(J_n^{\bullet,\bullet})\big)=H^{p+q}_{Z_n}(X,\mathcal{F}). 
\end{align}
Furthermore we have a natural morphism of double complexes
$$ \varprojlim_n J_n^{\bullet,\bullet} \rightarrow J_n^{\bullet,\bullet}.$$
As mentioned in EGA III, p. 30, a morphism of double complexes induces a natural morphism of spectral sequences (for both filtrations)
This implies we have a morphism $ {}^IE_1^{\bullet \bullet} \rightarrow {}^IE_1^{'\bullet \bullet}$
which is nothing but the morphism of complexes
$$ H^0_{Z}(X,\mathcal{I}^\bullet)={\varprojlim_n} H^0_{Z_n}(X,\mathcal{I}^\bullet) \hookrightarrow H^0_{Z_n}(X,\mathcal{I}^\bullet)$$
inducing the natural morphism $\varphi:H^*_Z(X,\mathcal{F})\rightarrow H^*_{Z_n}(X,\mathcal{F})$ in cohomology. The mentioned equality follows also by arguments from https://ivv5hpp.uni-muenster.de/u/pschnei/publ/pap/xsymm.pdf. Furthermore, the first spectral sequence implies $H^*_Z(X,\mathcal{F})=H^{*}(R\varprojlim_nK_n^\bullet).$
Then by the induced morphism on spectral sequences and the degeneration on the second page we have a commutative diagram
$\require{AMScd}$
\begin{CD}
 H^{n}(R\varprojlim_nK_n^\bullet) @>>> H^*_{Z_n}(X,\mathcal{F})\\
@V  V V= @VV  V\\
 {}^{II}E_2^{0n} @>>> {}^{II}E_2^{'0n}
\end{CD}
which is
$\require{AMScd}$
\begin{CD}
 H^*_Z(X,\mathcal{F})@>\varphi>> H^*_{Z_n}(X,\mathcal{F})\\
@V  g V V= @VV id  V\\
  {\varprojlim_n} H^*_{Z_n}(X,\mathcal{F})@>p_n>> H^*_{Z_n}(X,\mathcal{F})
\end{CD}
what we were looking for.
