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Let $C \in sSet$ be a quasicategory. $*$ denote the join of simplicial sets. Then consider $- * X \colon sSet \to sSet_{X/}$, where $sSet_{X/}$ is the slice category of simplicial set under $X$, with objects $X \to Y$.

For $p \colon X \to Y$, $X_{/p}$ is $p$ evaluated at right adjoint of $- * X$.

$C_{/x}$ is $x \colon \triangle^0 \to C$ evaluated at right adjoint of $-*\triangle^0$.

I'm trying to make sense of the bijective correspondence as shown below.

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I get that $\triangle^n \to C_{/x}$ correspond to $\triangle^n * \triangle^0 \to C$, but I have no idea how other maps correspond.

Any help would be appreciated, thanks!

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(This is adapted from Chapter 3 of "The Theory of Quasicategories and Its Applications Vol. 2" by Andre Joyal. Warning: The notation is different.)

Notation and Background

Fix a simplicial set $T \in \textsf{sSet}$. Let $T \downarrow \textsf{sSet}$ denote the category whose objects are simplicial maps $T \to X$ and whose morphisms are commuting triangles.

We have a functor $$ \left(-\right) \ast T \colon \textsf{sSet} \to T \downarrow \textsf{sSet}$$ that maps a simplicial set $X$ to the object $T = \emptyset \ast T \to X \ast T$ in the slice category. This functor admits a right adjoint $$ T \downarrow \left(-\right) \colon T \downarrow \textsf{sSet} \to \textsf{sSet}$$

Given an object $t \colon T \to X$ in $T \downarrow \textsf{sSet}$, the $n$-simplices of $T \downarrow t$ are given by: $$ \left(T \downarrow t\right)_{n} = \textsf{sSet}\left(\Delta^{n},T \downarrow t\right) = \left(T \downarrow \textsf{sSet}\right)\left(T \to \Delta^{n} \ast T, t \colon T \to X\right)$$. When the context is clear, we will denote objects $t \colon T \to X$ in the slice category $T \downarrow \textsf{sSet}$ by their codomain $X$. For instance, we write $T \downarrow X$ instead of $T \downarrow t$.

Now, consider a simplicial map $S \xrightarrow{j} T$. We have an induced natural transformation $$j \downarrow \left(-\right) \colon T \downarrow \left(-\right) \to S \downarrow \left(-\right) $$ On an object $t \colon T \to X$ in $T \downarrow \textsf{sSet}$, this is given on $n$-simplices by: $$ \left(j \downarrow X\right)_{n} \colon \left(T \downarrow X\right)_{n} \to \left(S \downarrow X\right)_{n} $$ $$ \left(\Delta^{n} \ast T \xrightarrow{x} X\right) \mapsto \left( \Delta^{n} \ast S \xrightarrow{\Delta^{n} \ast j} \Delta^{n} \ast T \xrightarrow{x} X\right) $$

Given simplicial maps $S \xrightarrow{j} T \xrightarrow{t} X \xrightarrow{f} Y$, the naturality square $\require{AMScd}$ \begin{CD} T \downarrow X @>j \downarrow X>> S \downarrow X \\ @V T \downarrow f VV @VV S \downarrow f V \\ T\downarrow Y @>>j \downarrow j> S \downarrow Y \end{CD} induces the pulbback slice $j \vartriangleright f \colon T \downarrow X \to \left(T \downarrow Y\right) \times_{S \downarrow Y} \left(S \downarrow X\right)$

Lemma

Given simplicial maps $i \colon A \to B$, $j \colon S \to T$, $t \colon T \to X$, and $f \colon X \to Y$, we have a 1-1 correspondence of lifting problems: $\require{AMScd}$ $$\begin{CD} \left(A \ast T\right) \cup_{A \ast S} \left(B \ast S\right) @> >> X \\ @V i \hat{\ast} j VV @VV f V \\ B \ast T @>> > Y \end{CD} \longleftrightarrow \begin{CD} A @> >> T \downarrow X \\ @V i VV @VV j \vartriangleright f V \\ B @>> > \left(T \downarrow Y\right) \times_{S \downarrow Y} \left(S \downarrow X\right) \end{CD} $$ Moreover, there is a 1-1 correspondence between their solutions. In particular, $$\left(i \hat{\ast} j\right) \pitchfork f \Leftrightarrow i \pitchfork \left(j \vartriangleright f\right)$$.

(The correspondence is given by repeatedly using the adjunction between join and slice. This is Lemma 3.14 in Joyal's "The Theory of Quasicategories and Its Applications Vol. 2".)

Answer to your question

$C_{/x}$ in your question corresponds to $\Delta^{0} \downarrow C$ in my notation. That is, the functor $\Delta^{0} \downarrow \left(-\right)$ applied to the object $x \colon \Delta^{0} \to C$.

Now, in the above lemma, take $i \colon A \to B$ to be the horn inclusion $v^{n}_{j} \colon \Lambda^{n}_{j} \hookrightarrow \Delta^{n}$, take $j \colon S \to T$ to be the inclusion $\emptyset \hookrightarrow \Delta^{0}$, $t \colon T \to X$ to be the 0-cell $x \colon \Delta^{0} \to C$, and $f \colon X \to Y$ to be unique map $C \xrightarrow{!} \Delta^{0}$.

Note that $$\left(\emptyset \hookrightarrow \Delta^{0}\right) \vartriangleright \left(C \xrightarrow{!} \Delta^{0}\right)$$ is precisely the map $\pi \colon C_{/x} \to C$ in your question.

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