$n$-lab says that that a limit (resp colimit) over the empty diagram in a category $\mathcal{C}$ is the terminal (resp initial) object of $\mathcal{C}$.
Is this true because if we are to give a morphism to each object in the empty diagram, we must factor through the limit over the empty diagram - where each object vacuuously has a morphism to the objects of the empty diagram, and hence necessarily has a unique morphism to this limit.
On
Given a small category $J$, a category $\mathcal C$ admits all limits of shape $J$ whenever the "constant diagram" functor $$ \mathcal C \to \mathcal C^J,\quad X\mapsto \{j \mapsto X, (i\overset f \to j) \mapsto \mathrm{id}_X\} $$ admits a right adjoint. When $J=\varnothing$, then $\mathcal C^J = 1$ (the terminal category). But a right adjoint of the unique functor $\mathcal C \to 1$ is precisely the selection of a terminal object in $\mathcal C$.
I only consider the limit case (colimits are dual).
The limit of a diagram $\Delta$ is a cone over $\Delta$ with a certain universal property (see https://en.wikipedia.org/wiki/Limit_(category_theory).
A cone over the empty diagram $\Delta_\emptyset$ is nothing else than an object $X$ of $\mathcal{C}$, and the universal property says that $X$ is a limit of $\Delta_\emptyset$ if and only if each object $Y$ admits a unique morphism $Y \to X$.