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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?

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    I've just seen this is also the proof in ncatlab: http://ncatlab.org/nlab/show/adjoint+functor2012-01-21
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    Dear Bruno: Here the symbol $\simeq$ means "is *canonically* isomorphic to", not just "is isomorphic to".2012-01-21
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    @Pierre: Yes, I'm aware of this, and in fact it is necessary for Yoneda lemma to apply. I thought it was only necessary for this... Hmm...2012-01-21
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    Detail: Here is a more direct link to the relevant part of the nLab entry: http://ncatlab.org/nlab/show/adjoint+functor#general_18.2012-01-21
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    Dear Bruno: If $D\to D_i$ is one of the natural maps, then all the isomorphisms in the chain are compatible with it.2012-01-21
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    @Pierre: I don't understand why...2012-01-21
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    Dear Bruno: I'll try to write a proof using the notation of [these notes](http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf) by Pierre Schapira. --- See more precisely Proposition 2.4.5 p. 36.2012-01-21
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    Dear Bruno, I posted an answer. Please feel free to tell me what you think of it.2012-01-21
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    Related: https://math.stackexchange.com/questions/271010/2018-11-28

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