TITLE: A proof that right adjoints preserve limits?
QUESTION [5 upvotes]: Assume that categories $\mathscr{B}$ and $\mathscr{C}$ have all limits of shape $\mathbf{J}$. Then there's a slick proof that if $G\colon \mathscr{C} \to \mathscr{B}$ is a right adjoint, $G \circ Lim_\mathscr{C} \cong Lim_\mathscr{B} \circ [\mathbf{J}, G]$. 
[$Lim_\mathscr{C} \colon [\mathbf{J}, \mathscr{C}] \to \mathscr{C}$ sends a diagram $D \colon \mathbf{J} \to \mathscr{C}$ to the vertex of a limit cone over $D$. $[\mathbf{J}, G]$ is a functor which, acting on objects in $[\mathbf{J}, \mathscr{C}]$ sends $D$ to $GD$. And the slick proof is to take left adjoints, noting that the left adjoint of $Lim$ is a constant functor, show very easily that the composite left adjoints are equal, so their right adjoints must be equal up to natural isomorphism. But proof details do not matter here.]
Now, I have seen it very briskly said -- e.g. in notes of a course by Peter Johnstone -- that this result shows that, at least when dealing with categories that have all limits of shape $\mathbf{J}$, right adjoints preserve those limits.
The usual definition of "$G$ preserves limits" is that if $(L, \pi_J)$ is a limit cone over $D$, then $(GL, G\pi_j)$ is a limit cone over $GD$.
But does the stated result establish that? One issue is that not every limit cone over $D$ has a vertex $Lim_\mathscr{C}D$ -- for the functor $Lim_\mathscr{C}$ has to be defined in terms of a choice from limit cones over $D$. So strictly speaking, the slick proof doesn't (as it stands, without augmentation) tell us what $G$ does to those other limit cones. [Added: OK: We can prove that if $G$ preserves one limit cone over $D$ it preserves all limit cones over $D$, and then make use of this fact here.] 
But more basically, even when we have a limit cone with a vertex $Lim_\mathscr{C}D$, what the result shows is that applying $G$ gives a result $X$ which a certain natural bijection maps to the vertex of a limit cone over $GD$. But it doesn't actually tell us that $X$ is the vertex of a limit cone over $GD$ as we require for full preservation. Or so it seems.
Given Peter Johnstone can do no wrong(!) where am I missing the point?

REPLY [2 votes]: I don't think that this proof shows the fully explicit mapping of any limit cone over $D$ to a limit cone over $GD$ that you want. That's a hard claim to justify precisely, so instead perhaps I can suggest how the slick proof could be refined to get exactly the claim you want. Let $J^l$ be the general cone over $J$, that is, $J$ with a disjoint initial object attached. Then we can refine the limit functor to a functor $\lim:[J,\mathcal{C}]\to [J^l,\mathcal{C}]$, with $\lim D$ simply a choice of limit cone over $D$. This limit functor is right adjoint to the restricton to $J$, and the same argument as that you alluded to shows that $G$ send the limit cone on $D$ specified by $\lim$ to something naturally isomorphic to the specified limit cone on $GD$, that is, to a limit cone. Then this composes with the natural isomorphism of any limit cone on $D$ to the specifed one to get the full result desired.