Lemma. 5.4.1.1 Lurie states that if $C$ is a simplicial category and $f_0:\partial \Delta^n \rightarrow N(C)$ is a map. $X=f_0(\{0\}), Y= f_0(\{n\})$, there is an induced map $$g_0:\partial (\Delta^1)^{n-1} \rightarrow Map_C(X,Y)$$
How is this map induced?
This comes from the description of the left adjoint to the homotopy coherent nerve, call it $\mathfrak C$. One has that $\mathfrak C\Delta^n(0,n)=(\Delta^1)^{n-1}$, while a computation with cocontinuity shows $\mathfrak C\partial \Delta^n(0,n)=\partial\left((\Delta^1)^{n-1}\right)$.