Definitions:

Recall the definition of the join of two simplicial sets. We may regard the functor $-\star Y$ as a functor $i_{Y,-\star Y}:Set_\Delta\to (Y\downarrow Set_\Delta)$ by replacing the resulting simplicial set $X\star Y$ with the canonical map $i_{Y,X\star Y}: Y\cong \emptyset\star Y \to X\star Y$. It is not hard to show that $i_{Y,-\star Y}$ commutes with colimits in $(Y\downarrow Set_\Delta)$, and therefore that it admits a right adjoint. We call the adjoint functor the *overcategory* or *over-simplicial-set* functor. We can give an explicit description of this simplicial set: Given an arrow $f:Y\to S$, define $(S\downarrow f)_n:=Hom_{(Y\downarrow Set_\Delta)}(i_{Y,\Delta^n\star Y},f)$.

As with all adjunctions, we have a unit and counit natural transformation $\eta_X:X\to (X\star Y\downarrow i_{Y,X\star Y})$, and $\epsilon_f: i_{Y,(X\downarrow f)\star Y}\to f$ respectively.

Then the question: Intuitively, the unit map should map $X$ to the simplicial subset of $(X\star Y \downarrow i_{Y,X\star Y})$ spanned by the "original" vertices of $X$, and I have seen this applied before (specifically in the case where $X$ is a simplex). However, I can't seem to figure out why this should be true formally. Is it true, and if so, how can we prove it?

Edit: As Tim mentions, the augmentation is given by the empty presheaf, that is, $\Delta^{-1}:=\emptyset$, which gives $X(-1):=Hom(\Delta^{-1}, X)=Hom(\emptyset,X)=\{\ast\}$