The category of presheaves $\mathrm{Pr}(\mathbf{C})$ is simply the category of functors from $\mathbb{C}^{\mathrm{op}}$ to the category of sets. The category of sheaves with respect to a topology $\mathcal{S}$ on $\mathbb{C}$ is denoted $\mathrm{Sh}(\mathbb{C}^{\mathrm{S}})$.

Say $u : \mathbb{C} \to \mathbb{D}$ is any functor. The pullback functor on presheaves will be written $u^{-1} : \mathrm{Pr}(\mathbb{D}) \to \mathrm{Pr}(\mathbb{C})$. It has left and right adjoints $u_!$ and $u_*$ respectively. It is shown in SGA4 [7] I.5.6 that the functor $u_!$ is fully faithful if and only if $u$ is fully faithful.

The category of sheaves with respect to the topology $\mathcal{S}$ on $\mathbb{C}$ will be denoted $\mathrm{Sh}(\mathbb{C}^{\mathrm{S}})$. We use $u_s$ to denote the composition $\mathrm{Sh}(\mathbb{C}^{\mathrm{S}}) \to \mathrm{Pr}(\mathbb{C}) \xrightarrow{u_!} \mathrm{Pr}(\mathbb{D}) \to \mathrm{Sh}(\mathbb{D}^{\mathrm{T}})$ where the first map is the fully faithful inclusion of sheaves into presheaves and the final map is sheafification. It is known (see [3]) that if $u$ is continuous that $u^{-1}$ preserves sheaves and by Lemma 7.14.3 of [2] that $u_s$ is a left adjoint to $u^{-1} : \mathrm{Sh}(\mathbb{D}^{\mathrm{T}}) \to \mathrm{Sh}(\mathbb{C}^{\mathrm{S}})$. From [4] and the cocontinuity we get a natural equivalence between $u^{-1} u_s$ and the identity and therefore we have the following

**LEMMA B.2.** — Suppose we are given a continuous, cocontinuous, fully faithful functor $u : \mathbb{C} \to \mathbb{D}$ between the categories underlying sub-canonical sites $\mathbb{C}^{\mathrm{S}}$ and $\mathbb{D}^{\mathrm{T}}$. Then the functor $u_s : \mathrm{Sh}(\mathbb{C}^{\mathrm{S}}) \to \mathrm{Sh}(\mathbb{D}^{\mathrm{T}})$ is fully faithful.

**Remark B.3.** — Notice that in the situation of Lemma B.2, $u_s$ commutes with colimits. Also by Lemma 7.14.5 of [2], $u_s$ preserves representable sheaves in the sense that $u_s(h_c) = h_{u(c)}$ for every $c \in \mathbb{C}$. We have

$$ (u^{-1} u_s(h_c))(c') = u_s(h_c)(u(c')) = h_{u(c)}(u(c')) = \mathrm{Hom}(u(c'), u(c)) $$

for every $c, c' \in \mathbb{C}$. Therefore since $u$ is fully faithful, we have natural isomorphisms $u^{-1} u_s(h_c) \cong h_c$ for every $c \in \mathbb{C}$.

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