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tim-at-topos committed May 15, 2024
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}
is a morphism in #{\mathcal{F}_{S'}} that is compatible with the canonical gluing data.
We thus obtain a \em{canonical functor} from the category #{\mathcal{F}_S} to the category of objects of #{\mathcal{F}_{S'}} endowed with gluing data with respect to the pair #{(\beta_1,\beta_2)}.
With this, we can also rephrase \ref{fga3.i-a.1-definition-1.3} by saying that \ref{fga3.i-a.1-definition-1.3-equation} is \em{#{\mathcal{F}}-exact} if the above functor is \em{fully faithful}, i.e. if the above functor defines an equivalence between the category #{\mathcal{F}_S} and a subcategory of the category of objects of #{\mathcal{F}_S} endowed with gluing data with respect to #{(\beta_1,\beta_2)}.
With this, we can also rephrase \ref{fga3.i-a.1-definition-1.3} by saying that \ref{fga3.i-a.1-definition-1.3-equation} is \em{#{\mathcal{F}}-exact} if the above functor is \em{fully faithful}, i.e. if the above functor defines an equivalence between the category #{\mathcal{F}_S} and a subcategory of the category of objects of #{\mathcal{F}_{S'}} endowed with gluing data with respect to #{(\beta_1,\beta_2)}.
}

\subtree[fga3.i-a.1-definition-1.5]{
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