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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 $\mathcal C$, then $j$ is in fact 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?

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?

I'm trying to understand J. P. Murre's 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 $\mathcal C$, $j$ is in fact 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?

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 $\mathcal C$, then $j$ is in fact 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?

I'm trying to understand J. P. Murre's 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 $\mathcal C$, $j$ is in fact 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 $\operatorname*{colim}_{(S, \tau) \in \mathcal I}\operatorname{Hom}_\mathcal C(S, X)$ 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?

I'm trying to understand J. P. Murre's 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 $\mathcal C$, $j$ is in fact 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?