In Lurie's book Higher Algebra, he makes the following definition:


**Definition 3.1.2.2:** Let $M^\otimes\to N(Fin\_\ast)\times\Delta^1$ be a correspondence from an $\infty$-operad $A^\otimes$ to another $\infty$-operad $B^\otimes$, let $q:C^\otimes\to O^\otimes$ be a fibration of $\infty$-operads and let $\overline{F}:M^\otimes\to C^\otimes$ be a map of generalized $\infty$-operads. We say that $\overline{F}$ is an operadic $q$-left Kan extension of $F\vert A^\otimes$ if the following condition is satisfied for every $b\in B^\otimes$:


($\ast$) Let $K=(M\_{act}^\otimes)\_{/b}\times\_{M^\otimes} A^\otimes.$ Then the composite map $$K^\vartriangleright\to (M^\otimes)\_{/b}^\vartriangleright\to M^\otimes\overset{\overline{F}}\to C^\otimes$$ is an operadic $q$-colimit diagram.


**My question is the following:**


What is the map $(M^\otimes)\_{/b}^\vartriangleright\to M^\otimes$? Does the cone point of the left hand side have a representative inside of $M^\otimes$? Does the existence of such a map implicitly require such a point to be in $M^\otimes$ already? 



Definition of Left Operadic Kan Extension for $\infty$-operads
Aaron's chat room remark is right: when restricted to $(M^\otimes)\_{/b}$ it is the canonical projection, and the cone point is sent to $b$.


Notice that for each object in the slice $f : x \to b$, there is a unique morphism from $f$ to the cone point in $(M^\otimes)\_{/b}^\vartriangleright$; this unique morphism is sent to $f$.


To describe the functor fully explicitly as a map of simplicial sets:


If an $n$-simplex $\sigma$ in $(M^\otimes)\_{/b}^\vartriangleright$ uses the cone point, the cone point comes at the end, so if the vertices of $\sigma$ are $v\_0, \ldots, v\_n$, there is some $k \le n$ such that the $v\_i$ with $i>k$ are the cone point and the $v\_i$ with $i \le k$ are not. Then the face on vertices $\{0, \ldots, k\}$ is a $k$-simplex in $(M^\otimes)\_{/b}$, and thus really a $(k+1)$-simplex $\tau$ in $M^\otimes$ with last vertex $b$. The required map is defined in two cases:


* if $k=n$, it sends $\sigma$ to $d\_{n+1}\tau$, the face of $\tau$ on $\{0, \ldots, n\}$, and
* if $k<n$, it sends $\sigma$ to $s\_{n-1} \cdots s\_{k+2} s\_{k+1} \tau$, the degenerate simplex on $\tau$ obtained by applying the last possible degeneracy $n-k-1$ times.


Also, the definition of this map works for $M^\otimes$ an arbitrary quasicategory and doesn't require it to be a correspondence between operads.


