@Dedalus It is not a morphism of sites, because the induced functor Sh(Y) → Sh(X) doesn't preserve finite limits.

@MarcHoyois Right! The problem is that the map of sites does not take products to products, right? Or is it something else causing this?

@Dedalus Stacks project has two candidates for morphism of sites, continuous functors and cocontinuous functors. The latter are covariant with the geometric morphism they induce, the former contravariant. It can't be a continuous functor of sites because it doesn't preserve products. It does seem to be cocontinuous but I think the associated morphism

Will be the usual one between etale topoi

@JoeBerner Morphism of site is different from continuous/cocontinuous functors. The functor Et/Y → Et/X is both continuous and cocontinuous, but not a morphism of sites (indeed because it doesn't preserve the final object). Its right adjoint Et/X → Et/Y is a morphism of site, inducing the geometric morphism f_*: Sh(Y) → Sh(X).

But f^*: Sh(X) → Sh(Y) is never a geometric morphism. If f is finite etale then f_* has a further right adjoint that's also a geometric morphism.

is there any software out there to help me visualize (and trace out differentials) in trigraded spectral sequences?

the indexing in these things confuses me to no end

i'm specifically interested in trigraded visualizations of the may spectral sequence and the algebraic novikov spectral sequence (at p=2, say)