Is the category of (co)cones a slice category? The definition of the category of cones reminds me of slice categories, except that we want an arrow between cones to induce (possibly) many commutative triangles.
Can $\mathsf{Cone}(F)$ be written as a slice category?
 A: Not really, but it is a subcategory of a slice category.
If $F : J → \mathscr C$ is a functor, then it is an object of the functor category $\mathscr C^J$. For every object $C$ of $\mathscr C$ there is a functor $ΔC : J → \mathscr C$, sending every object of $J$ to $C$, and every morphism to $\mathrm{id}_C$.
Natural transformations from $ΔC$ to $F$ are exactly the cones from $C$ to $F$.
So every object of $\operatorname{Cone} F$ is an object of the slice category $\mathscr C^J/F$.
Furthermore, $Δ$ extends to a functor $Δ : \mathscr C → \mathscr C^J$ (called the diagonal functor). If $f : C → D$ is a morphism in $\mathscr C$, then we define the natural transformation $Δf : ΔC → ΔD$ by $(Δf)_j = f$, for every object $j$ of $J$. This functor is faithful when $J$ is non-empty.
Now take two cones, say $γ : ΔC → F$ and $δ : ΔD → F$, and $f : C → D$ a morphism of cones. You can easily check that $Δf$ is then a morphism in $\mathscr C^J/F$. Since $Δ$ is faithful, you can take $\operatorname{Cone} F$ to be a subcategory of $\mathscr C/F$.
Or in other words, as Zhen Lin mentions in a comment, $\operatorname{Cone} F$ is (isomorphic to) the comma category $(Δ \downarrow F)$, and the limiting cone is the terminal object in that category. 
This btw. is a useful perspective to have because it follows by certain basic results on adjoint functors that if every functor $F : J → \mathscr C$ has a limit, then there exists a functor $\operatorname{lim} : \mathscr C^J → \mathscr C$ right adjoint to $Δ$, such that $\operatorname{lim} F$ is a limit of $F$ for every functor $F : J → \mathscr C$.
