Recall that for $H$ a (connected) algebraic group, we defined a profinite group $$ 1 \to \pi_1^\mathrm{c}(H) \to \tilde{H} \to H \to 1. $$ If $1 \to K \to H_1 \to H_2 \to 1$ is a short exact sequence, where $K$ is finite étale, then there is an induced short exact sequence $1 \to \pi_1^\mathrm{c}(H_1) \to \pi_1^\mathrm{c}(H_2) \to K \to 1$.
Lemma 1. If $1 \to H \to H_1 \to H_2 \to 1$ is a short exact sequence of connected linear algebraic groups, then we have a right exact sequence $$ \pi_1^\mathrm{c}(H) \to \pi_1^\mathrm{c}(H_1) \to \pi_1^\mathrm{c}(H_2) \to 1. $$
Proof.
For surjectivity of $\pi_1^\mathrm{c}(H_1) \to \pi_1^\mathrm{c}(H_2)$, we note that for any connected finite étale $H_2^\prime \to H_2$, the fiber product $$ H_1 \times_{H_2} H_2^\prime \to H_2^\prime $$ is connected because its kernel $H$ is connected and $H_2^\prime$ is also connected.
If now $\Lambda$ is a finite $\mathbb{Z}_ \ell$-algbera or some algebraic extensions of $\mathbb{Q}_ \ell$, then given any character $$ \chi \colon \pi_1^\mathrm{c} \to \Lambda^\times $$ we can compose with $\pi_1^\mathrm{et} \twoheadrightarrow \pi_1^\mathrm{c}$ and get a local system $\mathrm{Ch}_ \chi$ on $H$. But this also has the property that $\tilde{H}$ has a group structure. In particular, this is a character sheaf so that we have an isomorphism $$ m^\ast \mathrm{Ch}_ \chi \cong \mathrm{Ch}_ \chi \boxtimes \mathrm{Ch}_ \chi $$ on $H \times H$ (with $m$ being the multiplication), satisfying the cocycle condition on $H \times H \times H$.
Definition 2. A character sheaf (or character local systems) on a (not necessarily connected) algebraic group $H$ is a rank-1 local system $\mathscr{L}$ equipped with an isomorphism $m^\ast \mathscr{L} \cong \mathscr{L} \boxtimes \mathscr{L}$ satisfying the cocycle condition. We denote by $\mathsf{CS}(H, \Lambda)$ the groupoid of character sheaves on $H$ with coefficient $\Lambda$.
Remark 3. If $H$ is connected, then $\mathsf{CS}(H, \Lambda)$ is a discrete groupoid and is moreover an abelian group. In this case, being a character sheaf is a property, i.e., the isomorphism $m^\ast \mathscr{L} \cong \mathscr{L} \boxtimes \mathscr{L}$ is determined by $\mathscr{L}$, and moreover $$ \Hom_\mathrm{cts}(\pi_1^\mathrm{c}(H), \Lambda^\times) \cong \mathsf{CS}(H, \Lambda). $$
But we want to upgrade this. Let $\mathcal{R}_ {\pi_1^\mathrm{c}(H), \mathbb{G}_ m}$ be the moduli space over $\mathbb{Z}_ \ell$ classifying (strongly) continuous $\mathbb{G}_ m$-representations of $\pi_1^\mathrm{c}(H)$. That is, we have $$ \mathcal{R}_ {\pi_1^\mathrm{c}(H), \mathbb{G}_ m}(A) = \lbrace \rho \colon \pi_1^\mathrm{c}(H) \to A^\times \rbrace $$ where $\rho$ has to satisfy the condition that $A$ as a $\pi_1^\mathrm{c}(H)$-module is a union $A = \bigcup_i V_i$ where each $V_i$ is a $\pi_1^\mathrm{c}(H)$-submodule that is finite over $\mathbb{Z}_ \ell$, and each $\pi_1^\mathrm{c}(H) \to \Aut(V_i)$ is continuous in the usual sense.
Proposition 4. The functor $\mathcal{R}_ {\pi_1^\mathrm{c}(H), \mathbb{G}_ m}$ is represented by an ind-scheme that is ind-finite over $\mathbb{Z}_ \ell$.
The point here is that the pro-$\ell$-quotient of $\pi_1^\mathrm{c}(H)$ is topologically finitely generated. We can check this using the lemma before.
Example 5. We can compute that $$ \mathcal{R}_ {\mathbb{Z}_ \ell, \mathbb{G}_ m} = \bigcup_{Z \subseteq \mathbb{G}_ {m,\mathbb{Z}_ \ell}} Z $$ where the union is over $Z$ that are finite over $\mathbb{Z}_ \ell$, satisfying the property that $Z_ {\mathbb{F}_ \ell} \subseteq \mathbb{G}_ m^\wedge$.
Now this is an ind-scheme, so we can consider the category of coherent sheaves on it. Then there exists a fully faithful exact functor $$ \mathsf{Coh}(\mathcal{R}_ {\pi_1^\mathrm{c}(H), \mathbb{G}_ m})^\heartsuit \to \mathsf{Shv}(H)^\heartsuit; \quad \mathscr{O}_ \chi = (\Spec \Lambda \to \mathcal{R}_ {\pi_1^\mathrm{c}(H),\mathbb{G}_ m})_ \ast \mathscr{O} \mapsto \mathrm{Ch}_ \chi. $$ The image is the thick abelian subcategory of $\mathsf{Shv}(H)^\heartsuit$ generated by character sheaves. We denote this by $$ \mathsf{Shv}_ \mathrm{mon}(H)^{\omega,\heartsuit} \subseteq \mathsf{Shv}(H)^\heartsuit. $$
Remark 6. Another way of thinking about this is to identify this $\mathsf{Coh}(\mathcal{R}_ {\pi_1^\mathrm{c}(H),\mathbb{G}_ m})^\heartsuit$ as the category of continuous $\pi_1^\mathrm{c}(H)$-modules on finite $\mathbb{Z}_ \ell$-modules. Then this is just sent to the corresponding representation of $\pi_1^\mathrm{et}(H)$.
Lemma 7. Let $f \colon H_1 \to H_2$ be a surjective. If $\mathscr{F} \in \mathsf{Shv}(H_2)^\heartsuit$ satisfies the property that $f^\ast \mathscr{F} \in \mathsf{Shv}_ \mathrm{mon}(H_1)^{\omega,\heartsuit}$ then $\mathscr{F} \in \mathsf{Shv}_ \mathrm{mon}(H_2)^{\omega,\heartsuit}$.
Proof.
Because $f^\ast \mathscr{F}$ is locally constant, so is $\mathscr{F}$, and so it corresponds to some representation of $\pi_1^\mathrm{et}(H_2)$. We divide into two cases: when $\ker f$ is finite étale and when $\ker f$ is connected. In the first case, we just look at the diagram $$ \begin{CD} 1 @>>> \pi_1^\mathrm{et}(H_1) @>>> \pi_1^\mathrm{et}(H_2) @>>> \ker f @>>> 1 \br @. @VVV @VVV @| \br 1 @>>> \pi_1^\mathrm{c}(H_1) @>>> \pi_1^\mathrm{c}(H_2) @>>> \ker f @>>> 1, \end{CD} $$ and observe that because $\ker(\pi_1^\mathrm{et}(H_1) \to \pi_1^\mathrm{c}(H_1))$ acts trivially, so does $\ker(\pi_1^\mathrm{et}(H_2) \to \pi_1^\mathrm{c}(H_2))$.
In the second case, we use previous lemma and look at the diagram $$ \begin{CD} \pi_1^\mathrm{et}(H) @>>> \pi_1^\mathrm{et}(H_1) @>>> \pi_1^\mathrm{et}(H_2) @>>> 1 \br @VVV @VVV @VVV \br \pi_1^\mathrm{c}(H) @>>> \pi_1^\mathrm{c}(H_1) @>>> \pi_1^\mathrm{c}(H_2) @>>> 1 \end{CD} $$ where the bottom row is exact and the last map on the top row is surjective. Then we can check that the map on kernel is surjective.
Let us now fix a character $\chi \colon \pi_1^\mathrm{c}(H) \to \Lambda^\times$, which induces $\mathrm{Ch}_ \chi$. We let $$ \mathsf{Shv}_ {\chi\mathrm{-mon}}(H) \subseteq \mathsf{Shv}(H) $$ be the presentable stable full ∞-subcategory generated by $\mathrm{Ch}_ \chi$. We then let $$ \mathsf{Shv}_ \mathrm{mon}(H) \subseteq \mathsf{Shv}(H) $$ be the category generated by $\mathrm{Ch}_ \chi$ over all $\chi$. We can check that $$ \mathsf{Shv}_ \mathrm{mon}(H) = \operatorname{Ind}(\mathsf{Shv}_ \mathrm{mon}(H)^\omega), \quad \mathsf{Shv}_ \mathrm{mon}(H)^\omega = \lbrace \mathscr{F} \in \mathsf{Shv}(H)^\omega : \mathcal{H}^i \mathscr{F} \in \mathsf{Shv}_ \mathrm{mon}(H)^{\omega,\heartsuit} \rbrace. $$ All of the functors $$ \mathsf{Mod}_ \Lambda \to \mathsf{Shv}_ {\chi\mathrm{-mon}}(H) \hookrightarrow \mathsf{Shv}_ \mathrm{mon}(H) \hookrightarrow \mathsf{Shv}(H) $$ admit continuous right adjoints.
Proposition 8. The right adjoints $\mathrm{Av}^\mathrm{mon} \colon \mathsf{Shv}(H) \to \mathsf{Shv}_ \mathrm{mon}(H)$ and $\mathrm{Av}^{\chi\mathrm{-mon}} \colon \mathsf{Shv}(H) \to \mathsf{Shv}_ {\chi\mathrm{-mon}}(H)$ are monoidal functors.
This is really why we wanted to pass to these big categories. To have a unit object, we need to have things that are not compact.