TITLE: Right adjoints preserve limits
QUESTION [29 upvotes]: In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an adjunction, then if $D:I\to \mathcal{D}$ is a diagram that has a limit, we have, for every $A\in \mathcal{C}$,
$\begin{align*} \hom_\mathcal{C} (A, G(\varprojlim D)) &\simeq \hom_{\mathcal{D}} (F(A),\varprojlim D)\\ & \simeq \varprojlim \hom_{\mathcal{D}}(F(A),D)\\& \simeq \varprojlim \hom_{\mathcal{C}}(A,GD) \\& \simeq \hom_{\mathcal{C}}(A,\varprojlim GD)\end{align*}$
because representables preserve limits. Whence, by Yoneda lemma, $G(\varprojlim D)\simeq \varprojlim GD$.
This is very slick, but I can't really see why the proof is finished. Yes, we proved that the two objects are isomorphic, but a limit is not just an object... Don't we need to prove that the isomorphism also respects the natural maps? That is,
if $\varphi:G(\varprojlim D)\to \varprojlim GD$ is the isomorphism, and $\alpha_i: \varprojlim D \to D_i$, $\beta_i:\varprojlim GD \to GD_i$ are the canonical maps for all $i\in I$, do we have that $\beta_i\varphi=G(\alpha_i)$?
I don't see how this follows from Awodey's proof. How can we deduce it?

REPLY [2 votes]: A new answer was added recently, and from my perspective, none of the answers reflect what I think the proof is trying to get at, so I thought I'd add another answer. 
However, I do agree with Zhen Lin's answer (+1) that the proof goes wrong when it assumes already that the limit of $GD$ exists. The point is rather to show that $(G\lim D,G(\alpha_i))$ form a limiting cone if $(\lim D,\alpha_i)$ form a limiting cone for $D$.
Other answers argue that there are better proofs, and I won't necessarily agree or disagree, but the idea of this proof is sound, so let me see if I can salvage it.
Review of the Yoneda Lemma
The key point is that part of the Yoneda lemma says that natural transformations from a Yoneda functor $\newcommand\C{\mathcal{C}}\newcommand\D{\mathcal{D}}\C(-,X)$ to another functor $F:\C^{\text{op}}\to \newcommand\Set{\mathbf{Set}}\Set$ are naturally isomorphic to the elements of $F(X)$.
In particular, if you have a natural isomorphism 
$$\C(-,L)\simeq \text{cones from $-$ to }D,$$
the natural isomorphism corresponds to a cone from $L$ to $D$, given by the image of $1_L\in\C(L,L)$, which is the limiting cone.
The argument
Thus, the way this argument should go is the following:
Let $L=\lim D$ for notational convenience.
$$
\begin{align}
\C(-,GL)&\simeq \D(F-,L) \\
&\simeq \lim_i \D(F-,D_i)\\
&\simeq \lim_i \D(-,GD_i)\\
&\simeq \text{cones from $-$ to } GD.
\end{align}
$$
This natural isomorphism proves that $GL$ is the limit of $GD$ already. To see what the limiting cone is, we have to see where $1_{GL}$ maps under the natural isomorphism.
Under the first natural isomorphism, $1_{GL}$ maps to the [counit] of the adjunction,
$\epsilon_L : FGL\to L$. The next natural isomorphism sends $\epsilon_L$ to 
$(\alpha_i\circ \epsilon_L)_i$, which belong to the limit of hom sets because the $(\alpha_i)$ do. Finally, the adjunct of 
$\alpha_i\circ \epsilon_L$ is given by 
$$G(\alpha_i\circ \epsilon_L)\circ \eta_{GL} = G\alpha_i \circ G\epsilon_L \circ \eta_{GL}
=G\alpha_i,$$
with the last equality being the triangle identities.
Finally, the last isomorphism is just a reinterpretation of the elements of the limit as cones, since checking what it means to take the limit in $\Set$ of the hom-sets against what it means for elements to be cones we see that they agree. Thus $(G\alpha_i)$ is the limiting cone corresponding to this natural isomorphism, as desired.
TL;DR
The natural isomorphism already encodes the data of the limiting cone. All we need to do is check that we get the expected limiting cone.