ct.category theory - Computing the image of the unit map of the join/overcategory adjunction for simplicial sets

Definitions:

Recall the definition of the join of two simplicial sets. We may regard the functor $-star Y$ as a functor $i_{Y,-star Y}:Set_Deltato (Ydownarrow Set_Delta)$ by replacing the resulting simplicial set $Xstar Y$ with the canonical map $i_{Y,Xstar Y}: Ycong emptysetstar Y to Xstar Y$. It is not hard to show that $i_{Y,-star Y}$ commutes with colimits in $(Ydownarrow Set_Delta)$, 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 $f:Yto S$, define $(Sdownarrow f)_n:=Hom_{(Ydownarrow Set_Delta)}(i_{Y,Delta^nstar Y},f)$.

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

Then the question: Intuitively, the unit map should map $X$ to the simplicial subset of $(Xstar Y downarrow i_{Y,Xstar Y})$ 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 $X(-1):=Hom(Delta^{-1}, X)=Hom(emptyset,X)={ast}$