Given a local system $\mathcal L$. Given a loop $\gamma:I\rightarrow X$, choose a sequence $\{U_i\}$ of small open sets that cover it where on each $U_i$, we have $\mathcal L|_{U_i}$ constant, therefore we can construct isomorphisms $\mathcal L_{a_i}\cong \mathcal L_{a_{i+1}}$ for a sequence of points $\{a_i\}$ which eventually loops back to the base point. Composing them we get an automorphism of $\mathcal L_{x}\cong k^n$. One can show that this construction only depends on the homotopy class of $\gamma$. Thus, we constructed $\rho:\pi_1(X,x)\rightarrow \mathrm{Aut}(\mathcal L_x)$. Conversely, given $\rho:\pi_1(X,x)\rightarrow\mathrm{Aut}_k(V)$ for some finite dimensional $k$-vector space $V$. Choose universal cover $\widetilde{X}$ and form the vector bundle

$$E_\rho=(\widetilde{X}\times V)/\pi_1(X,x)\longrightarrow X$$

where one mods out by diagonal action by $\pi_1$. We then can define the local system

$$\mathcal L(U)=\{\mathrm{locally\ constant\ sections\ }U\rightarrow E_\rho\}$$

i.e. sections that locally of the form $U\xrightarrow{u\mapsto (u,v)}E_\rho|_{U}=U\times V$ for some $v$. It is not hard to check that this is a local system which is mutually inverse to the previous construction.

We will show it for $\Lambda=\mathbb Z/\ell^n\mathbb Z$, and the rest can be obtained from taking inverse limits and tensoring. Let $\mathcal F$ be a lisse $\mathbb Z/\ell^n\mathbb Z$-sheaf, and let $\mathrm{Et}(\mathcal F)\rightarrow X$ be its etale space. The fiber $\mathcal F_{\overline{x}}$ is the fiber of this finite etale cover above $\overline{x}$, so the functorial action of $\pi_1^{\mathrm{\acute et}}(X,\overline{x})$ on fibers gives a continuous representation

$$ \rho_{\mathcal F}:\pi_1^{\mathrm{\acute et}}(X,\overline{x})\rightarrow \mathrm{Aut}_{\mathbb Z/\ell^n\mathbb Z}(\mathcal F_{\overline{x}}). $$

Conversely, suppose $\rho:\pi_1^{\mathrm{\acute et}}(X,\overline{x})\rightarrow\mathrm{Aut}_{\mathbb Z/\ell^n\mathbb Z}(M)$ is a continuous representation on a finite $\mathbb Z/\ell^n\mathbb Z$-module $M$. For each etale $U\rightarrow X$, define the (etale) sheaf of $\mathbb Z/\ell^n$-modules associated to $\rho$

$$\mathcal F_{\rho}(U)=\mathrm{Hom}_{\pi_1^{\mathrm{\acute et}}(X,\overline{x})\textrm{-Set}}(U_{\overline{x}},M)$$

with $\mathbb Z/\ell^n$-module structure inherited from $M$ pointwise. We claim that $\mathcal F_\rho$ is lisse. Since $M$ is finite and $\rho$ is continuous, $\ker(\rho)\subseteq \pi_1^{\mathrm{\acute et}}(X,\overline{x})$ is open. Choose a connected finite etale cover $Y\rightarrow X$ corresponding to an open subgroup $H\subseteq\ker(\rho)$, and choose a geometric point $\overline{y}\rightarrow Y$ above $\overline{x}$. Then $ \pi_1^{\mathrm{\acute et}}(Y,\overline{y})=H $ acts trivially on $M$. Hence the pullback of $\mathcal F_\rho$ to $Y_{\mathrm{\acute et}}$ is the sheaf associated to the trivial representation of $\pi_1^{\mathrm{\acute et}}(Y,\overline{y})$ on $M$, which is the constant sheaf $\underline{M}_Y$. Thus $\mathcal F_\rho$ becomes constant after the finite etale cover $Y\rightarrow X$. Since $M$ is finite, $\mathcal F_\rho$ is locally constant constructible, hence lisse.

The two constructions are mutually inverse. Starting with $\rho$ on $M$, the stalk of $\mathcal F_\rho$ at $\overline{x}$ is

$$ (\mathcal F_\rho)_{\overline{x}}\cong M, $$

because the geometric fiber of the object $\overline{x}\rightarrow X$ is the one-point $\pi_1^{\mathrm{\acute et}}(X,\overline{x})$-set, so evaluating an equivariant map on that point recovers an element of $M$. Under this identification, the monodromy action on $(\mathcal F_\rho)_{\overline{x}}$ is exactly the original action $\rho$, since the action on the fiber is the action used in the equivariance condition defining $\mathcal F_\rho$. Conversely, start with a lisse sheaf $\mathcal F$ and let $M=\mathcal F_{\overline{x}}$ with its monodromy representation $\rho_{\mathcal F}$. For every etale $U\rightarrow X$, a section $s\in\mathcal F(U)$ determines a map on geometric fibers

$$ U_{\overline{x}}\longrightarrow M $$

sending a geometric point $\overline{u}\rightarrow U$ above $\overline{x}$ to the value of $s$ in the stalk $\mathcal F_{\overline{u}}\cong \mathcal F_{\overline{x}}$, where the identification is given by monodromy. This map is $\pi_1^{\mathrm{\acute et}}(X,\overline{x})$-equivariant. Hence we get a natural map

$$ \mathcal F(U)\longrightarrow \mathrm{Hom}_{\pi_1^{\mathrm{\acute et}}(X,\overline{x})\textrm{-Set}}(U_{\overline{x}},\mathcal F_{\overline{x}}). $$

After pulling back to an etale cover on which $\mathcal F$ is constant, this map is visibly an isomorphism: both sides are locally constant functions to the same finite module $M$. Therefore it is an isomorphism after etale descent, and hence for all $U$. Thus the sheaf reconstructed from $\rho_{\mathcal F}$ is canonically isomorphic to $\mathcal F$.

For a local system $\mathcal L$, send it to the vector bundle $\mathcal L\otimes_{\mathbb C_X}\mathcal O_X$, equipped with the flat connection

$$\mathcal L\otimes_{\mathbb C_X}\mathcal O_X\xrightarrow{1\otimes \mathrm d}\mathcal L\otimes_{\mathbb C_X} \Omega^1_X$$

where $\mathrm d:\mathcal O_X\rightarrow\Omega^1_X$ is the differential. Conversely, send a pair $(\mathcal E,\nabla)$ to the sheaf of horizontal sections $\mathrm{Ker}(\nabla)$. We claim this is a local system. For a small polydisc $U\subseteq X$ trivializing $\mathcal E$ with $\mathcal E|_U=(\mathcal O_X|_U)^r$, the connection has form $\nabla|_U=\mathrm d+A$ where $A\in \mathrm M_{r}(\Omega^1_X)$. For a local section $s:U\rightarrow \mathbb C^r$, the equation $\nabla(s)=0$ determines a system of first order linear PDEs.

$$\frac{\partial s}{\partial z_k}+A_k s=0 \tag{1}$$

for $k=1,\dots,n$, where $\{z_k\}$ is a local frame and $A=\sum_{k}A_k\mathrm dz_k$. From flatness $\nabla^2=0$, we get $\mathrm{d}A+A\wedge A=0$, i.e.

$$\frac{\partial A_j}{\partial z_i}-\frac{\partial A_i}{\partial z_j}+\left[A_i, A_j\right]=0$$

This is equivalent to $D_i=\partial_i+A_i$ commuting $D_iD_j=D_jD_i$. Holomorphic Frobenius theorem implies that the PDE in equation $(1)$ is locally solvable with any initial value, meaning for each $v\in \mathcal E_x$, there exists unique holomorphic local section $s_v:U\rightarrow\mathbb C^r$ satisfying equation (1) with $s_v(x)=v$. It is not hard to check that the two constructions are mutually inverse.

See ⟦cite:HTT08⟧ .