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?

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