If $\mathcal{C}$ is a category and $\lambda:\Delta_D \to id_{\mathcal{C}}$ is a cone for the identity functor, and $F:J \to \mathcal{C}$ is a functor such that $F\lambda:\Delta_D \to F$ is a limiting cone, then it follows that $D$ is an initial object of $\mathcal{C}$. This is Lemma 1 on page 230 in Maclane's Categories for the Working Mathematician. The proof given however is pretty down to earth and does not generalize immediately to higher category theory. Is there a more highbrowed way of proving this that works in a more general context than 1categories?

The definition of object being initial is obviously equivalent to the statement "the inclusion of onepoint category is final". Here by final functor I mean $p: I\to C$, such that for any $F: C\to D$ we have $$\mathrm{Lim}\ Fp \simeq \mathrm{Lim}\ F$$ The criterion of being final is proved in MacLane 9.3 (there such functors are called "initial", and "final" is used for colimits). Then $p: \ast \to C$ is the initial object iff $$\mathrm{Lim}\ 1_C \simeq \mathrm{Lim}\ 1_C\circ p = \mathrm{Lim} p = p$$ This is the theorem you mention. In higher category theory the concepts of limits and final functors work just the same, so we have a similar theorem. Note that the criterion of finalness differs: all corresponding overcategories must be contractible. 

