(m-1-1) -- node {$f$} (m-1-2);
        \draw[right hook->] (m-1-2) -- (m-1-3);
        \draw[right hook->] (m-1-1) -- (m-2-1);
        \draw[->,dashed] (m-2-1) -- node[swap] {$\tilde f$} (m-1-3);
        \draw[->,dashed] (m-2-1) -- node {$\tilde f'$} (m-1-2);
    \end{tikzpicture}$$ such that $f\rvert_{\{0\}}=t_\infty$. Then by [@Lurie:2009un Proposition 2.4.4.9.], the arrow $\tilde f$ exists (making the diagram commute), but since ${\operatorname}{Fun}_{K^{op}}^{{\operatorname}{Cart}}({K^{op}},X)$ is the full subcategory of ${\operatorname}{Fun}_{K^{op}}({K^{op}},X)$ spanned by the Cartesian sections, and all the vertices of $\tilde f$ lie in the image of $f$, and hence in ${\operatorname}{Fun}_{K^{op}}^{{\operatorname}{Cart}}({K^{op}},X)$, it follows that $\tilde f$ factors through a map $\tilde f':\Delta^n\to {\operatorname}{Fun}_{K^{op}}^{{\operatorname}{Cart}}({K^{op}},X)$.

Thus $t_\infty\in \lim p'$ is an initial object.

By construction, for every $k\in K$, $\pi_k(t_\infty)\in p'(k)$ is the image of the initial object $t_\infty(k)\in X_k$ under the equivalence of $\infty$-categories $X_k\xrightarrow{\cong} p'(k)$. Thus (2) follows from the uniqueness of initial objects.

Now, we can interpret arbitrary colimits in terms of initial objects using the concept of an undercategory, as follows: Suppose that ${\mathcal}{C}\in {{\mathscr{C}\mathrm{at}}}_\infty$ is an $\infty$-category and $q:I\to {\mathcal}{C}$ is a diagram of shape $I$ (where $I$ is a small simplicial set). Then a colimit diagram for $q$ is equivalent to an initial object of the undercategory, ${\mathcal}{C}^{q/}$.

Suppose that $K$ is a (small) simplicial set and $\bar p':K^{\lhd}\to 