Lemma 99.3.6. Let $p : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The following are equivalent

1. the diagonal $\Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X}$ is limit preserving, and

2. for every directed limit $U = \mathop{\mathrm{lim}}\nolimits U_ i$ of affine schemes over $S$ the functor

$\mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}$

is fully faithful.

In particular, if $p$ is limit preserving, then $\Delta$ is too.

Proof. Let $U = \mathop{\mathrm{lim}}\nolimits U_ i$ be a directed limit of affine schemes over $S$. We claim that the functor

$\mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i}$

is fully faithful if and only if the functor

$\mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \longrightarrow \mathcal{X}_ U \times _{(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U} \mathop{\mathrm{colim}}\nolimits (\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i}$

is an equivalence. This will prove the lemma. Since $(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_ U = \mathcal{X}_ U \times _{\mathcal{Y}_ U} \mathcal{X}_ U$ and $(\mathcal{X} \times _\mathcal {Y} \mathcal{X})_{U_ i} = \mathcal{X}_{U_ i} \times _{\mathcal{Y}_{U_ i}} \mathcal{X}_{U_ i}$ this is a purely category theoretic assertion which we discuss in the next paragraph.

Let $\mathcal{I}$ be a filtered index category. Let $(\mathcal{C}_ i)$ and $(\mathcal{D}_ i)$ be systems of groupoids over $\mathcal{I}$. Let $p : (\mathcal{C}_ i) \to (\mathcal{D}_ i)$ be a map of systems of groupoids over $\mathcal{I}$. Suppose we have a functor $p : \mathcal{C} \to \mathcal{D}$ of groupoids and functors $f : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \to \mathcal{C}$ and $g : \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \to \mathcal{D}$ fitting into a commutative diagram

$\xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i \ar[r]^ g & \mathcal{D} }$

Then we claim that

$A : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _\mathcal {D} \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i$

is fully faithful if and only if the functor

$B : \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} \mathop{\mathrm{colim}}\nolimits (\mathcal{C}_ i \times _{\mathcal{D}_ i} \mathcal{C}_ i)$

is an equivalence. Set $\mathcal{C}' = \mathop{\mathrm{colim}}\nolimits \mathcal{C}_ i$ and $\mathcal{D}' = \mathop{\mathrm{colim}}\nolimits \mathcal{D}_ i$. Since $2$-fibre products commute with filtered colimits we see that $A$ and $B$ become the functors

$A' : \mathcal{C}' \to \mathcal{C} \times _\mathcal {D} \mathcal{D}' \quad \text{and}\quad B' : \mathcal{C}' \longrightarrow \mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}')$

Thus it suffices to prove that if

$\xymatrix{ \mathcal{C}' \ar[d]_ p \ar[r]_ f & \mathcal{C} \ar[d]^ p \\ \mathcal{D}' \ar[r]^ g & \mathcal{D} }$

is a commutative diagram of groupoids, then $A'$ is fully faithful if and only if $B'$ is an equivalence. This follows from Categories, Lemma 4.34.9 (with trivial, i.e., punctual, base category) because

$\mathcal{C} \times _{\Delta , \mathcal{C} \times _\mathcal {D} \mathcal{C}, f \times _ g f} (\mathcal{C}' \times _{\mathcal{D}'} \mathcal{C}') = \mathcal{C}' \times _{A', \mathcal{C} \times _\mathcal {D} \mathcal{D}', A'} \mathcal{C}'$

This finishes the proof. $\square$

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