A known theorem in category theory is
Suppose $\mathscr{C}$ has an initial object $c$. Then $c$, along with its unique maps, forms the limit of the identity functor $\mathscr{C} \to \mathscr{C}$.
It seems like the converse should be true, i.e. if $c$ is a limit of Id then $c$ is initial, but I can't seem to prove it. Can anyone help?
Let $(π_c : i → c)_c$ be the limiting cone to $\mathrm{Id}$, and $f : i → c$ a morphism. You want to prove that $f = π_c$. Since $π$ is a cone, we have $π_c = fπ_i$, so it suffices to prove that $π_i = \mathrm{id}$. To do this, use the fact that $π_c$'s are jointly monic: $π_cπ_i = π_c = π_c ∘ \mathrm{id}$ for all $c$, so $π_c = \mathrm{id}$.