\[
  \varphi_0 : \mathcal{U}(X) \longrightarrow \{0, 1\}
\]

\[
  \varphi_0(U) = 1 \quad \text{sss} \quad x \in U ,
\]

\[
  \varphi(U) \neq \emptyset \iff x \in U .
\]

\[
  \varphi(F) \longrightarrow \varphi(U)
\]

\[
  (\ast) \qquad \varinjlim_{(Y, \xi) \in C_\varphi^{\circ}} \check{Y}
  \longrightarrow \varphi
\]

\[
  \mathrm{Hom}\bigl(\varinjlim_{C_\varphi} \check{Y}, \varphi\bigr) \simeq
  \varprojlim_{C_\varphi} \mathrm{Hom}(\check{Y}, \varphi) \simeq
  \varprojlim_{C_\varphi} \varphi(\widetilde{Y}) .
\]

\[
  (R_0^{i}) \leftleftarrows (R_1^{ij})
\]

\begin{align*}
  \underline{\underline{\mathrm{Quot}}}_{F/X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Quot}}(F'/X'/S')\bigr)
     = \mathrm{Quot}(F'/X'/S') \\
  \underline{\underline{\mathrm{Hilb}}}_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Hilb}}(X'/S')\bigr)
     = \mathrm{Hilb}(X'/S')
     \qquad = \underline{\underline{\mathrm{Quot}}}_{\mathcal{O}_X/X/S}(S') \\
  \underline{\underline{\mathrm{Div}}}_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Div}}(X'/S')\bigr)
     = \mathrm{Div}(X'/S')
\end{align*}

\begin{align*}
  \underline{\underline{\mathrm{Hom}}}_S(X, Y)(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Hom}}_{S'}(X', Y')\bigr)
     = \mathrm{Hom}_{S'}(X', Y') = \mathrm{Hom}_S(X \times_S S', Y) \\
  \underline{\underline{\mathrm{Isom}}}_S(X, Y)(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Isom}}_{S'}(X', Y')\bigr)
     = \mathrm{Isom}_{S'}(X', Y') \\
  \underline{\underline{\mathrm{Aut}}}_S(X)(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Aut}}_{S'}(X')\bigr)
     = \mathrm{Aut}_{S'}(X') \\
  \underline{\underline{\mathrm{Imm}}}_S(X, Y)(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Imm}}_{S'}(X', Y')\bigr)
     = \mathrm{Imm}_{S'}(X', Y')
\end{align*}

\[
  \Bigl(\underline{\underline{\prod}}_{Y/S} X/Y\Bigr)(S')
  = \Gamma\bigl(S', \underline{\Gamma}_{Y'/S'}(X'/Y')\bigr)
  = \bigl[\Gamma_{Y'/S'}(X'/Y') = \bigr]\;\Gamma(X'/Y')
\]

\[
  \underline{\underline{\Gamma}}_{Y/S}(X/Y)
  = \underline{\underline{H}}^{0}_{Y/S}(X/Y)
\]

\begin{align*}
  \underline{\underline{\mathrm{Prépic}}}_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Prépic}}_{X'/S'}\bigr)
     = \mathrm{Prépic}(X'/S')
     \qquad = H^1(X', \mathcal{O}_{X'}^{*}) \\
  \underline{\underline{\mathrm{Prépic}}}'_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Prépic}}'_{X'/S'}\bigr)
     = \mathrm{Prépic}'(X'/S')
     \qquad = \Gamma\bigl(S', \underline{H}^{1}_{X'/S'}(\underline{\mathcal{O}}^{*}_{X'})\bigr) \\
  \underline{\underline{\mathrm{Pic}}}_{X/S}(S')
    &= \Gamma\bigl(S', \underline{\mathrm{Pic}}_{X'/S'}\bigr)
     = \mathrm{Pic}(X'/S')
     \qquad = \widetilde{\underline{\underline{\mathrm{Prépic}}}}_{X/S}(S')
\end{align*}

\begin{align*}
  \mathrm{Hom}\Bigl(\underline{\underline{\prod}}_{X/S} \underline{F}, \underline{M}\Bigr)
    &= H^0\bigl(\underline{F} \otimes_S \underline{M}\bigr), &
  \underline{\mathrm{Hom}}\Bigl(\underline{\prod}_{X/S} \underline{F}, \underline{M}\Bigr)
    &= \underline{H}^{0}_{X/S}\bigl(\underline{F} \otimes_S \underline{M}\bigr) \\
  \underline{V}\Bigl(\prod_{X/S} \underline{F}\Bigr)
    &= \prod_{X/S} V(\underline{F}) &
  \underline{V}\Bigl(\underline{\underline{\prod}}_{X/S} \underline{F}\Bigr)
    &= \underline{\underline{\mathrm{Hom}}}_{X/S}(F, \underline{\mathcal{O}}_X)
\end{align*}

\begin{align*}
  \mathrm{Hom}\bigl(\underline{\underline{\mathrm{Hom}}}_{X/S}(F, G), \underline{M}\bigr)
    &= \mathrm{Hom}_{\mathcal{O}_X}(F, G \otimes_{\mathcal{O}_S} M) \\
  \mathrm{Hom}_{\mathcal{O}_{S'}}\bigl(\underline{\underline{\mathrm{Hom}}}_{X/S}(F, G) \otimes_{\mathcal{O}_S} S', \mathcal{O}_{S'}\bigr)
    &= \mathrm{Hom}_{\mathcal{O}_{X'}}(F', G') \\
  \underline{\mathrm{Hom}}_{\mathcal{O}_S}\bigl(\underline{\underline{\mathrm{Hom}}}_{X/S}(F, G), \underline{M}\bigr)
    &= \underline{\mathrm{Hom}}_{X/S}(F, G \otimes M)
\end{align*}