This is a subentry of the entry differential cohomology.
We discuss here how the general theory of differential nonabelian cohomology developed there relates to the standard theory of abelian differential cohomology.
The crucial input for translating the general setup to the familar abelian differential cohomology setup is
the “naturality of the differential Quillen adjunction” as discussed at path ∞-groupoid which gives commutative diagram
and their version for the infinitesimal path ∞-groupoid?
the deRham theorem for ∞-Lie groupoids? which for $A = \mathbf{B}^n \mathbb{R}$ identifies
$Y \to A_{flat}^{inf}$ with a degree $n$ real cococyle;
$\Pi^{inf}(Y) \to A$ with the corresponding deRham cocycle.
Ordinary abelian differential cohomology starts (following Hopkins-Singer) with specifying real cohomology classes
on an abelian $A$, then pulling these back along a given $A$-cocycle $g : Y \to A$ to the composite
and then identifying them there by a homotopy $h$ with a differential form