Right adjoints preserve limits

The proof is very nice, but one needs to be absolutely clear about what needs to be proven in order to understand it. The claim is, if $\lambda_i : \varprojlim D \to D_i$ is a limiting cone in $\mathcal{D}$, then $G \lambda_i : G(\varprojlim D) \to G D_i$ is a limiting cone in $\mathcal{C}$. We do not postulate the existence of $\varprojlim G D$; this is what we are going to prove.

So suppose we are given a cone $\mu_i : X \to G D_i$ in $\mathcal{C}$. This yields a cone of hom-sets $$(\mu_i)_* : \mathcal{C}(A, X) \to \mathcal{C}(A, G D_i)$$ and since $\varprojlim \mathcal{C}(A, G D_i) \cong \mathcal{C}(A, G(\varprojlim D))$ naturally in $A$ (by the argument you cited), by the Yoneda lemma it follows that there is a unique natural transformation $\varphi_* : \mathcal{C}(-, X) \Rightarrow \mathcal{C}(-, G(\varprojlim D))$ such that $$(\mu_i)_* = (G \lambda_i)_* \circ \varphi_*$$ where $\varphi_*$ comes from a morphism $\varphi : X \to G(\varprojlim D)$. Thus $G \lambda_i : G(\varprojlim D) \to G D_i$ is indeed a limiting cone.

This is essentially taken from the proof of Proposition 2.4.5 p. 36 in these notes by Pierre Schapira.

Let me use the abbreviation $$ L:=\lim_{\underset{i}{\longleftarrow}}\quad. $$

Let $C$ and $D$ be categories, let $b:I^{op}\to C$, $i\mapsto b_i$, be a projective system, let $Lb$ be its limit (we assume it exists), let $F:C\to D$ be a functor, let $G:D\to C$ be its left adjoint (we assume it exists), and let $y$ be an object of $D$.

We have the following commuting squares of morphisms and isomorphisms, where the vertical arrows are induced by the $i$ th canonical projections: $$ \begin{matrix} D(y,FLb)&\simeq&C(Gy,Lb)\\ \downarrow&&\downarrow\\ D(y,Fb_i)&\simeq&C(Gy,b_i), \end{matrix} $$

$$ \begin{matrix} C(Gy,Lb)&\simeq&LC(Gy,b)\\ \downarrow&&\downarrow\\ C(Gy,b_i)&=&C(Gy,b_i), \end{matrix} $$

$$ \begin{matrix} LC(Gy,b)&\simeq&LD(y,Fb)\\ \downarrow&&\downarrow\\ C(Gy,b_i)&\simeq&D(y,Fb_i), \end{matrix} $$

$$ \begin{matrix} LD(y,Fb)&\simeq&D(y,LFb)\\ \downarrow&&\downarrow\\ D(y,Fb_i)&=&D(y,Fb_i). \end{matrix} $$ By splicing these squares, we get the following commuting square of morphisms and isomorphisms: $$ \begin{matrix} D(y,FLb)&\simeq&D(y,LFb)\\ \downarrow&&\downarrow\\ D(y,Fb_i)&=&D(y,Fb_i), \end{matrix} $$ which is what we wanted.

EDIT A. Let's go back to the first square: $$ \begin{matrix} D(y,FLb)&\simeq&C(Gy,Lb)\\ \downarrow&&\downarrow\\ D(y,Fb_i)&\simeq&C(Gy,b_i). \end{matrix} $$ The isomorphisms are given by the adjunction. If $p_i:Lb\to b_i$ denotes the $i$ th canonical projection, then the first downward arrow is $D(y,Fp_i)$, and the second is $C(Gy,p_i)$. To show that the square commutes, we only need to invoke the fact that the adjunction is functorial in the second variable.

Now to the second square: $$ \begin{matrix} C(Gy,Lb)&\simeq&LC(Gy,b)\\ \downarrow&&\downarrow\\ C(Gy,b_i)&=&C(Gy,b_i). \end{matrix} $$ By assumption, we have chosen a representing object $Lb$ and an isomorphism $$ C(x,Lb)\simeq LC(x,b) $$ functorial in $x\in\text{Ob}(C)$. We have a natural map from $LC(x,b)$ to $C(x,b_i)$ --- because $LC(x,b)$ is a projective limit of sets. Then we define the map from $C(x,Lb)$ to $C(x,b_i)$ as the one which makes the above square commutative. All this being functorial in $x$, the Yoneda Lemma yields the morphism $p_i:Lb\to b_i$ used above.

EDIT B. The third and fourth squares are handled similarly. So we end up with the square $$ \begin{matrix} D(y,FLb)&\simeq&D(y,LFb)\\ \downarrow&&\downarrow\\ D(y,Fb_i)&=&D(y,Fb_i), \end{matrix} $$ which commutes for all $i$. What we want to prove is the existence of an isomorphism $FLb\simeq LFb$ such that the square $$ \begin{matrix} FLb&\simeq&LFb\\ \downarrow&&\downarrow\\ Fb_i&=&Fb_i \end{matrix} $$ commutes for all $i$. But, in view of Yoneda, the above square commutes because the previous one does.

EDIT C. Alternative wording of the poof that $$ \begin{matrix} C(x,Lb)&\simeq&LC(x,b)\\ \downarrow&&\downarrow\\ C(x,b_i)&=&C(x,b_i) \end{matrix} $$ commutes:

A morphism $f\in C(x,Lb)$ is given by a family $f_\bullet=(f_j)_{j\in I}\in LC(x,b)$ satisfying the obvious compatibility conditions, and we have $f_j=p_j\circ f$ for all $j$. So, $f$ and $f_\bullet$ correspond under the isomorphism in the above square. Moreover, the first vertical arrow maps $f$ to $f_i$, and the second vertical arrow maps $f_\bullet$ to $f_i$.

Awodey sometimes hand waves his proofs, especially the ones in which you have to get your hands dirty--for instance, his proof that every presheaf is a colimit of representables.

Anyway, in this case, application of Yoneda is overkill because it is no harder to prove RAPL just by using adjointness:

Suppose $\left \langle L,\lambda _{i} \right \rangle$ is a limit cone to $D$. Then $\left \langle GL,G\lambda _{i} \right \rangle$ is a cone to $GD$. Suppose $\left \langle X,\mu _{i} \right \rangle$ is a cone to $GD$. Then taking adjoints, we get a cone to $D$, namely, $\left \langle FX,\mu _{i}^{*}\right \rangle$. Since $\left \langle L,\lambda _{i} \right \rangle$ is a limit cone, we get an arrow $\phi ^{*}:FX\rightarrow L$ unique with property that $\lambda _{i}\circ \phi ^{*}=\mu ^{*}_{i}$. Taking adjoints again, it is easy to see that $\phi :X\rightarrow GL$ is the unique arrow making the required triangle commute, for

if we write \begin{array}{ccc} \hom( X,GL) & \rightarrow & \hom (FX,L) \\ \downarrow & & \downarrow \\ \hom(X,GD _{i}) & \rightarrow & \hom(FX,D_{1} ) \end{array} and follow $\phi $ around the square, we see that $\lambda _{i}\circ \phi ^{*} =(G\lambda _{i}\circ \phi )^{*}$. But LHS of this is just $\mu ^{*}_{i}$, so that $G\lambda _{i}\circ \phi =\mu_{i} $ and $\phi $ is unique because $\phi ^{*}$ is.

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.