Refined hexagons for differential cohomology

Cheeger-Simons differential characters and differential $K$-theory are refinements of ordinary cohomology theory and topological $K$-theory respectively, and they are examples of differential cohomology. Each of these differential cohomology theories fits into a hexagon on the cohomology level. We show that these differential cohomology theories fit into hexagons on the cocycle level, and the hexagons on the cocycle level induce the hexagons on the cohomology level.