Lemma 7.28.2. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a functor. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$. Set $U = u(V)$. Assume that

1. $\mathcal{C}$ and $\mathcal{D}$ have all finite limits,

2. $u$ is continuous, and

3. $u$ commutes with finite limits.

There exists a commutative diagram of morphisms of sites

$\xymatrix{ \mathcal{C}/U \ar[r]_{j_ U} \ar[d]_{f'} & \mathcal{C} \ar[d]^ f \\ \mathcal{D}/V \ar[r]^{j_ V} & \mathcal{D} }$

where the right vertical arrow corresponds to $u$, the left vertical arrow corresponds to the functor $u' : \mathcal{D}/V \to \mathcal{C}/U$, $V'/V \mapsto u(V')/u(V)$ and the horizontal arrows correspond to the functors $\mathcal{C} \to \mathcal{C}/U$, $X \mapsto X \times U$ and $\mathcal{D} \to \mathcal{D}/V$, $Y \mapsto Y \times V$ as in Lemma 7.27.2. Moreover, the associated diagram of morphisms of topoi is equal to the diagram of Lemma 7.28.1. In particular we have $f'_*j_ U^{-1} = j_ V^{-1}f_*$.

Proof. Note that $u$ satisfies the assumptions of Proposition 7.14.7 and hence induces a morphism of sites $f : \mathcal{C} \to \mathcal{D}$ by that proposition. It is clear that $u$ induces a functor $u'$ as indicated. It is clear that this functor also satisfies the assumptions of Proposition 7.14.7. Hence we get a morphism of sites $f' : \mathcal{C}/U \to \mathcal{D}/V$. The diagram commutes by our definition of composition of morphisms of sites (see Definition 7.14.5) and because

$u(Y \times V) = u(Y) \times u(V) = u(Y) \times U$

which shows that the diagram of categories and functors opposite to the diagram of the lemma commutes. $\square$

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