For simplicity here only the affine (affinoid) case is written here but the framework work with general case.
Definition
A unipotent isocrystal on \(\overline{A}\) is an \(A^\dagger_K\)-module \(M\) and an integrable connection
\[ \nabla : M \to M\otimes_{A^\dagger_K} (\Omega^1_{A^\dagger} \otimes K) \]
obtained by iterated extensions of trivial connections \((A^\dagger_K, d)\). In this case the extension is actually a free module.
A morphism of unipotent isocrystals is a morphism of \(A^\dagger\)-modules \(f:M\to N\) such that \(\nabla_N \circ f = (f\otimes \Omega_K) \circ \nabla_M\).
The category of unipotent isocrystals on \(\overline{A}\) is denoted by \(\mathcal{U}(\overline{A})\), which as a category only depends on \(\overline{A}\) independent of the choice of lift.
Example
Any rank two unipotent isocrystal \(M\) is an extension in \(\mathrm{Ext}^1_{\mathcal{U}}(1,1)\). This splits due to freeness (non-canonical), so it is isomorphic to some \((\mathcal{O}^2, \nabla)\). Let’s see what happens inside, writting out
If we denote the injection \(\mathcal{O}_X\to M\) giving out the first frame section \(e_1\), and choose a non-canonical section that inverts the surjection \(M\to \mathcal{O}_X\) which maps \(e_2\), then the vertical arrows says
\(\nabla (e_1) = 0\) because the injection commutative diagram
\(\nabla(e_2) = 0\) when you mod out \(e_1\), so \(\nabla(e_2) = \omega e_1\) for some \(\omega\).
The requirement that \(\nabla\) is a flat connection gives \(\nabla(\omega e_1) = d\omega \wedge e_1=0\), so \(\omega\) is a closed form.
We have found out that \(\nabla = d\cdot I_2 + \Lambda\) where
\[ \Lambda = \begin{pmatrix} 0 & \omega \\ 0 & 0 \end{pmatrix} \]
In fact this correspondence gives a bijection
\[ \mathrm{Ext}^1_{\mathcal{U}(\overline{A})}(1, 1) \cong H^1_{MW}(\overline{A}/K),\quad [\Lambda] \mapsto [\omega]. \]
In Detail
Let’s consider why the above morphism \(\varphi\) is an isomorphism in detail. This means
It is well-defined, i.e. isomorphic unipotent isocrystals give the same cohomology class.
It is surjective, luckily this is obvious because \(\omega\) can be any closed form.
It is injective, i.e. if \(\omega=\,\mathrm{d}f\) is exact, then the corresponding \(\Lambda\) gives a connection that splits, it is isomorphic to a connection whose matrix is semisimple.
Tensor Structure
The category \(\mathcal{U}(\overline{A})\) is a rigid abelian tensor category, and it can be made into a Tannakian category with the fibre functors. Let \(x\in X_\kappa(\kappa)\) be a rational point in reduction, the fibre functor at \(x\) is a functor from \(\mathcal{U}(\overline{A})\) into \(K\)-vector spaces of flat sections over a local residue disk \(U_x\) of \(x\),
\[ \omega_x : \mathcal{U}(\overline{A})\to \mathrm{Vec}_K, \quad \omega_x[(M, \nabla)] := \{ v\in M(U_x), \nabla v = 0 \}. \]
Overconvergence is required to make \(\dim M(U_x) = \mathrm{rank}M\). Finding horizontal sections for a unipotent connection is the same as iterated integratoin, which is feasible because of overconvergence.
\(\omega_x\) can also be viewed as a pullback \(x^*\) to an isocrystal on \(\mathrm{Spec}(\kappa)\).
The category \(\mathcal{U}(\overline{A})\) together with fibre functor \(\omega_x\) determines an affine proalgebraic fundamental group
\[ G = G_x = \pi_1(\mathcal{U}(\overline{A}), \omega_x) \]
whose functor of points over any \(K\)-algebra \(F\), \(G(F)\) is given by
\[G(F) = \mathrm{Aut}_{\otimes}(\omega_x \otimes F),\]
here \(\omega_x \otimes F\) is the functor \((\otimes F)\circ \omega_x\), The group \(\mathrm{Aut}_{\otimes}(\omega_x\otimes F)\) is the group of natural automorphisms for the functor \(\alpha: \omega_x \otimes F \to \omega_x \otimes F\) that commutes with the tensor structure, i.e. \(\alpha_{M\otimes N} = \alpha_M \otimes \alpha_N\) and \(\alpha_{\mathcal{O}} = \mathrm{id}\).
There is an equivalence of categories (similar to Riemann-Hilbert correspondence) of finite dimensional \(K\)-representations of \(G\) and unipotent isocrystals on \(\overline{A}\)
\[ \mathrm{Rep}_K(\pi_1(\mathcal{U}(\overline{A}),\omega_x)) \simeq \mathcal{U}(\overline{A}) \]
Reference
Besser, Heidelberg Lectures on Coleman Integration