Why can we take the colimit over the category of elements?

Question

I'm trying to understand J. P. Murre's Tata notes on Grothendieck's theory of the fundamental group. For a Galois category $\mathcal C$ (which I'm taking to be locally small) with fundamental functor $F : \mathcal C \to \mathsf{FinSet}$, Murre shows that $F$ is pro-representable by showing that $F(X)$ is naturally isomorphic to $\operatorname*{colim}_{(S, \tau) \in \mathcal I^\mathrm{op}}\operatorname{Hom}_\mathcal C(S, X)$, where $\mathcal I$ is the full subcategory of the category of elements $\mathcal E$ consisting of minimal pairs. A pair $(S, \tau)$ is minimal if whenever $$ j : (X, \xi) \to (S, \tau) $$ is a morphism in $\mathcal E$ with $j : X \hookrightarrow S$ monic in $\mathcal C$, in fact $j$ is an isomorphism. My concern is that, although Murre calls $\mathcal E$ a set, it seems like $\mathcal I$ might not be a small category if $\mathcal C$ itself is not small, so I don't see how we know that $\operatorname*{colim}_{(S, \tau) \in \mathcal I^\mathrm{op}}\operatorname{Hom}_\mathcal C(S, X)$ exists.

I was told that the category of étale coverings over a fixed locally Noetherian connected prescheme is (essentially?) small, and if that's true, then I could believe that the categories of finite separable extensions of a fixed field and covering spaces over a fixed connected space (the categories I want to show are Galois for a presentation I need to give) are also (essentially) small, so should I just assume that a Galois category is (essentially) small? Alternatively, is there an argument that shows $\mathcal I^\mathrm{op}$ is cofinally small, perhaps using the fact that $\mathcal E^\mathrm{op}$ is filtered?

Answer 1

If I'm not mistaken, a cosequence of the arguments in Murre's proof is that the inclusion $J^{\text{op}}\colon\mathcal I^{\text{op}}\hookrightarrow \mathcal E ^{\text{op}}$ is a final functor: It's easy to see that if $H\colon \mathcal A\hookrightarrow \mathcal B$ is fully faithful and $\mathcal B$ is filtered, then $H$ is final if and only if for any $b\in\mathcal B$ there exists $a\in\mathcal A$ and a map $b\to Ha$ in $\mathcal B$. In your case this condition is satisfied by point (*) of the proof.

Now use the property of final functors applied to $J^{\text{op}}$: this says (in particular) that if the colimit of $G\colon \mathcal E^{\text{op}}\to \mathcal K$ exists, then also the colimit of $GJ^{\text{op}}\colon \mathcal I^{\text{op}}\to\mathcal K$ exists, and in that case they coincide. In your situation we already know that the colimit indexed on $\mathcal E^{\text{op}}$ exists and is equal to $F$ (it is always true, independently of the size of $\mathcal C$, that any functor $F\colon \mathcal C\to \text{Set}$ is the colimit of the diagram indexed on the category of elements), thus you can conclude.