The unit of the total décalage adjunction

We consider the décalage construction \({{\,\mathrm{Dec}\,}}\) and its right adjoint \(T\). These functors are induced on the category of simplicial objects valued in any bicomplete category \({\mathcal {C}}\) by the ordinal sum. We identify \(T{{\,\mathrm{Dec}\,}}X\) with the path object \(X^{\Delta [1]}\) for any simplicial object X. We then use this formula to produce an explicit retracting homotopy for the unit \(X\rightarrow T{{\,\mathrm{Dec}\,}}X\) of the adjunction \(({{\,\mathrm{Dec}\,}},T)\). When \({\mathcal {C}}\) is a category of objects of an algebraic nature, we then show that the unit is a weak equivalence of simplicial objects in \({\mathcal {C}}\).