For $C$ an ordinary category and $c \in C$ an object of $C$, the ordinary over category $C\downarrow c$ satisfies the universal property that for any other category $C'$ there is a natural equivalence of categories
where
$C' \star [0]$ denotes the category $C'$ with a freely adjoined terminal object $0$;
$Hom_{c}(C' \star [0], C)$ denotes the category of pairs $(F,\gamma)$, where $F: C' \star [0]\to C$ is a functor and $\gamma:F(0)\to c$ is an isomorphism in $C$.
The object $c$ can be seen as a functor $c: [0]\to C$. From this point of view, $\gamma$ is a natural transformation from $F\circ\iota$ to $c$, where $\iota: [0]\to C' \star [0]$ is the inclusion functor.
Remaining in the classical setting, even more is true: $C'\star [0]$ is a particular case of the join operation between categories, which admits a terse description in terms of the cograph of a profunctor.
More precisely, let $C,D$ be two categories; their join $C\star D$ is defined to be the category whose set of objects is the disjoint union of the sets of objects of $C$ and $D$, and where we add one and only one morphism between any $c\in C$ and any other $d\in D$. This is precisely the cograph of $C\uplus_\omega D$ of the unique profunctor $\omega\colon C^\text{op}\times D\to {\rm Set}$ sending any two $(c,d)$ to the singleton.
It is extremely easy, with this definition, to show that given a functor $p\colon K\to C$, the category $C_{/p}$ of cones over $p$ (and the category $C_{p/}$ of co-cones) are uniquely characterized by the following universal properties:
where $\text{Fun}^p(D\star K, C)$ denotes the category whose objects are functors which coincide with $p$ when restricted to the full subcategory $K\subset K\star D$ (see HTT Prop. 1.2.9.2). This characterize $C_{/p}$ as a representative for the functor $D\mapsto \text{Fun}_p(K\star D,C)$.
The idea of the definition of over category in the context of quasi-categories is to mimic this universal property. This relies crucially on generalizing the construction $C' \star [0]$ to the context of quasi-categories, in terms of the join of quasi-categories.
Fosco Loregian Warning: this paragraph is highly conjectural and for the moment I’m not able to offer any proof for these statements. In particular I would like to show * The equivalence between the quasicategorial and the simplicially enriched definition; * Does the “classical” definition of join between simplicial sets $K\star S$ coincide with the coherent nerve of the simplicial category $C[K]\star C[S]$? This would automatically entail that the coherent nerve in $\star$-monoidal, as claimed in HTT 1.2.8.2.
A word about the quasicategorical join operation: Joyal’s Prop. 3.1 suggests to interpret the join operation of simplicial sets as the convolution of presheaves. Nevertheless, it seems that the “pro-functorial” definition has something to say even in the $(\infty,1)$-categorical case: instead of the quasicategorical model, we want to consider the simplicially enriched model for $(\infty,1)$-categories. In this setting, the join of two $(\infty,1)$-categories $C,D\in\text{sSet-Cat}$ can easily be interpreted as the cograph $C\uplus_\omega D$ of the terminal $\text{sSet}$-profunctor sending any two objects $(C,D)$ to the terminal simplicial set.
See also this question on MO.
Therefore, let $F : K \to C$ be a morphism of quasi-categories; the over-quasi-category $C_{/F}$ is the quasi-category characterized by the property that for any other quasi-category $S$ there is a natural equivalence of quasi-categories
where
$S \star K$ is the join of quasi-categories of $S$ with $K$;
$Hom_{F}( S \star K, C )$ is the quasi-category of pairs $(f,\gamma)$, where $f : S \star K \to C$ is a morphism of quasi-categories and $\gamma\colon f\circ \iota\to F$ is an isomorphism and $\iota:K\to S \star K$ is the natural inclusion.
Here one sees the advantage of having worked with the full quasi-categories of morphisms rather than with Hom-Sets. Indeed, if $[0]$ denotes the terminal quasi-category, then any quasi-category $C$ is naturally equivalent to $Hom([0],C)$.
Therefore the above description of $C_{/F}$ is reduced to the following
Let $F : K \to C$ be a morphism of quasi-categories. The over-quasi-category $C_{/F}$ is the quasi-category $Hom_{F}( [0] \star K, C )$.
See proposition 1.2.9.2, p. 44 and the text leading to and including proposition 1.2.9.3 of
See also chapter 3 (_Join and Slices_) in
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