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In the book of Artin and Mazur, there is an appendix to pro-categories, specifically in A.2.2 one can find the following:

It can be shown ([25]) that morphisms between the functors associated to pro-objects X,Y are in 1-1 correspondence with elements of Hom(X,Y)

In the bibliography [25] is Quillen's Homotopical Algebra.

Question: I am searching for that proof in Quillen's work, but I can't find it. Someone can help?

Edit: As John suggested, I exlpain in details what is the question.

Let $C$ be a category, we can associate to $C$ a pro-category of pro-objects (see pro-object). Let $X$ be a pro-object indexed by $I$, then we have a a functor $C \rightarrow Set$ which to $T \in Ob_C \mapsto \varprojlim_I Hom(X_i, T)$, on morphisms it is defined in the natural way. Let this functor be $hX$.

The question is: It can be shown ([25]) that morphisms between the functors associated to pro-objects $X,Y$ are in 1-1 correspondence with elements of $Hom(X,Y)$ as pro-objects.

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    $\begingroup$ Could you include a precise statement of the result you want a reference for? I imagine not everyone reading your question is within arms reach of Artin--Mazur. $\endgroup$
    – John Pardon
    Commented Mar 13, 2018 at 2:10

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I am almost certain that that is a typo for [15] which is the key initial text on pro-categories.

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  • $\begingroup$ Thank you. The Exposé in question is the 195th. Unfortunately, Grothendieck doesn't give a proof of this fact. $\endgroup$
    – HaroldF
    Commented Mar 13, 2018 at 11:18
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    $\begingroup$ There are descriptions in several places in the literature, I think. In the book I wrote (with Jean-Marc Cordier) on Shape Theory (Dover, for the most accessible edition), page 43 has a discussion of that. Given the description of limits and colimits where the indexing category is (co)filtering, it is not that difficult to work out from the definitions. You may find other discussions of this in other texts on shape theory, but I happen to have a copy of my version at hand! $\endgroup$
    – Tim Porter
    Commented Mar 13, 2018 at 11:40
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What you're asking about is the equivalence between two standard definitions of $Pro(C)$. I've just added an explanation of the equivalence to the nlab page.

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