In SGAIV.1, Exp. III, sec. 5, sites $(C,J)$ are localised with respect to a presheaf $X$ on $C$ (not nec. just a representable one).

**Question1:** There it is asserted that if $F \to X$ is a morphism of presheaves on C, i.e., an object of $PreSh(C)/X$, then its image under the equivalence $PreSh(C)/X \to PreSh(hC/X)$ is a sheaf iff F is isomorphic to the cartesian product ${X}x_{ia(X)}ia(F)$ considered over $X$ in $PreSh(C)/X$. Has this been worked out somewhere? While working out the details one encounters some points more difficult than the material in Exp.'s II and III preceding it.

**Question2:** If G is a sheaf on $(hC/X)$ and $i_X$ denotes the inclusion of $Sh(hC/X)$ into $PreSh(hC/X)$, is ${j_X}_!(i_X(G))$ always a sheaf on $C$? (The topology $J$ is not necessarily subcanonical.)

Some of the details of this section can be extended to a general pair of adjoint functors and their categorical localisation to slice categories, which leads one to pose the following:

**Question3:** Has the discussion there been extended to arbitrary pairs of adjoint functors (L,R), $L:C \to C'$, $R:C' \to C$, (L',R'), $L':D \to D'$, $R':D' \to D$ and an equivalence of categories (A,B), $A:C \to D, B:D \to C$ (perhaps with R, R' fully faithful)?