Adjunction on simplicial sets I I am trying to understand the proof adjunction described in page 244-245, of Joyals Theory of Quasicategories.

Background: $$i^*:S/I \rightarrow S/\partial I = S \times S$$
  
  
*
  
*$S$ is category of simplicial sets. 
  
*$I$ is the simplicial set $1 \star 1$, where $1$ is terminal object $\star$ is join operation (explained also in the notes).  $\partial I= 1 \sqcup 1$. 
  
*$S/I$ is over category. Objects are $X \rightarrow I$, $X \in S$. 
  
*The construction is as follows, we given $X \rightarrow I$ in $S/I$, we $i^*X$, is the pullback of $X$ along inclusion $\partial I \rightarrow I$. 
  


In proof Prop. 3.5, pg 245, the author assumes the existence of right adjoint $i_*$. How is this true? 
 A: The category $S$ is locally cartesian closed, as is any presheaf category.
Indeed, for any $A\in S$, the category $S/A$ is equivalent to the category of presheaves over $\Delta/A$ (the category of elements of $A$). Any map $f:A\to B$ induces a functor $\Delta/f: \Delta / A \to \Delta / B$ and you can check that the restriction functor
$$ {-} \circ (\Delta/f) : \widehat{\Delta/B} \to \widehat{\Delta/A}$$ 
fits in a commutative square as follows:
$$\require{AMScd} 
\begin{CD}
\widehat{\Delta/B} @>{{-} \circ (\Delta/f)}>> \widehat{\Delta/A}
\\
@V\simeq VV @VV\simeq V 
\\
S/B @>>{f^\ast}> S/A
\end{CD}
$$
where the two verticals maps are the equivalence mentioned above. As the top map admits a right adjoint (the right Kan extension functor), so does the bottom map. All the constructions are explicit, so you can derive a formula for the right adjoint $f_\ast$ if needed.
Alternatively you can follow the recipe of showing that exponential object exist in $S$ (this is after all the special case of $B=1$ above) and adapt it to directly describe $f_\ast$.
