Limits in category of cones. I'm trying to do exercise 2.17.2 in Borceux's "Handbook of Categorical Algebra":
Consider a functor $F: \mathfrak{D} \to \mathfrak{C}$ and the category of cones on $F$. Show that $F$ has a limit if and only if the functor $U$ from the category of cones on $F$ to $\mathfrak{C}$ mapping a cone to its vertex has a colimit.
Here is what I thought, however I'm not sure if I should take a different approach or just fill the gaps in the proof.  I hope you can help me:
$\Rightarrow$ If $F$ has a limit $L$ we know that  limit is a terminal object in the category of cones on $F$ .So the inclusion functor is a final functor. And we get the result.
$\Leftarrow$ Now suppose $U$ has a colimit $L$, now we fix some $D \in \mathfrak{D}$ and there exists morphisms from each cone vertex to $FD$.
This makes $FD$ the vertex of a cocone $\implies \exists ! ~ \alpha_D: L \to FD$.
So $(L, \alpha_D)$ is a cone such that for every other cone $C_i, ~ \exists ~ h_i: C_i \to L$.  How do I prove uniqueness of the factorization?
PS: I edited the first implication because I found a proposition that helped me. So i just need to finish the last one.
 A: The conceptual key is that a terminal object in a category is the same thing as a colimit for the whole category (i.e. a colimit for the identity functor on the category as a diagram in that category).  
To see this note the mere existence of a co-cone from a whole category to one of its objects $T$ shows $T$ is weakly terminal and the co-cone component $c_T:T\rightarrow T$ coequalizes any pair of arrows $f,g:A\rightarrow T$ in the category.  But then $c_T$ induces a co-cone map from the  co-cone to itself, so the colimit property says it is the identity on $T$.  So there cannot be distinct arrows $f\neq g:A\rightarrow T$ in the category. 
As you say, a limit for $F$ is a terminal object in the category of cones on $F$.  So it is a colimit for the identity diagram in that category.  So you need to show a colimit for the identity diagram in the cone category amounts to the same thing as a colimit to the diagram of vertices of those cones in $\mathfrak{C}$.  That follows quickly from the reasoning you have done, showing a colimit $L$ for the diagram of vertices is the vertex of a cone $(L, \alpha_D)$ over $F$.
