I am trying to understand computation of the adjunction described in page 245, of Joyals Theory of Quasicategories. This is a follow up to my other post.
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 $\partial I \rightarrow I$.
My question: how does one compute,
$$i^* (\Lambda^n_k \rightarrow I)$$
I am hoping there is a simple expression for the pullback. Hopefully of the form of horns and simplices.