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.