  We consider the d\'ecalage construction $\operatorname{Dec}$ 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
$T \operatorname{Dec}X$ 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 $X\to T\operatorname{Dec}X$ of the adjunction
$(\operatorname{Dec},T)$. 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 $\mathcal{C}$.
