When I say "$\infty$-category", I mean a simplicial set such that all inner horns have not necessarily unique fillers. In Lurie's Higher Topos Theory there's the following
Definition. Given an $\infty$-category $\mathcal{C}$, a simplicial set $L$, a map of simplicial sets $p : L \to \mathcal{C}$, there's a unique $\infty$-category $\mathcal{C}_{/p}$, the over-category, such that, for every simplicial set $K$, we have $$\operatorname{Hom}_{\mathbf{sSet}}(K,\mathcal{C}_{/p}) \cong \operatorname{Hom}_p(K \star L,\mathcal{C}),$$ where $K \star L$ means the join of the simplicial sets $K$ and $L$, and where the right-hand side means all morphisms $K \star L \to \mathcal{C}$ commuting with the inclusion morphism $L \to K \star L$ and $p : L \to \mathcal{C}$.
It's then said that this "can be dualized", leading to the notion of an under-category; no further details are given. I've tried working them out myself, but it does not make sense to me.
Attempt 1. Given an $\infty$-category $\mathcal{C}$, a simplicial set $L$, a map of simplicial sets $p : \mathcal{C} \to L$, there's a unique $\infty$-category $\mathcal{C}_{p/}$ such that, for every simplicial set $K$, we have $$\operatorname{Hom}_{\mathbf{sSet}}(K,\mathcal{C}_{p/}) \cong \operatorname{Hom}_p(L \star K,\mathcal{C}),$$ where the right-hand side refers to morphisms $L \star K \to \mathcal{C}$ commuting with the maps $L \star K \to L$ and $p : \mathcal{C} \to L$. This can't be it, because there's no such thing as a map $L \star K \to L$.
Attempt 2. Given an $\infty$-category $\mathcal{C}$, a simplicial set $L$, a map of simplicial sets $p : \mathcal{C} \to L$, there's a (unique) $\infty$-category $\mathcal{C}_{p/}$ such that, for every simplicial set $K$, we have $$\operatorname{Hom}_{\mathbf{sSet}}(C_{p/},K) \cong \operatorname{Hom}_p(\mathcal{C},L \star K),$$ where the right-hand side refers to morphisms $\mathcal{C} \to L \star K$ commuting with the maps $p : \mathcal{C} \to L$ and the inclusion map $L \to L \star K$. This can't be it, bcause there's only one such map in the right-hand side, namely the composition.
It's probably something really obvious I'm missing, but for now I have the following
Question. What is an under-category?
Attempt 1 is the right isomorphism, but you dualized too much. $Hom_p(L\star K, \mathcal C)$ still means the maps commuting with the inclusion of $L$ and the map $p:L\to \mathcal C$. To help see this, consider the case that $L$ is terminal and $\mathcal C$ is a category-you should recover the slice category of the objects under $p$.