> **1 Preliminaries.**

> **1.1 Functoriality of Presheaves.**

Let $\phi\colon R\longrightarrow S$ be a morphism of rings. There is a natural functor
$$f\colon\mathbf{Alg}_S\longrightarrow\mathbf{Alg}_R$$
taking an $S$-algebra $S\rightarrow A$ to the $R$-algebra $R\rightarrow S\rightarrow A$.

As remarked in [this nLab page](https://ncatlab.org/nlab/show/functoriality+of+categories+of+presheaves) (and developed in detail in SGA IV, [Exposé I](http://www.normalesup.org/~forgogozo/SGA4/01/01.pdf), Section 5), there exists an induced adjoint triple of functors between the corresponding presheaf categories:


[![adjunction][1]][1]

where $f^*\colon\mathrm{PSh}(\mathbf{Alg}_R)\longrightarrow\mathrm{PSh}(\mathbf{Alg}_S)$ is given by precomposition with $f$.

> **1.2 Relation to Schemes.**

We may restrict $f^*$ to the full subcategory $\mathbf{Aff}_R$ of $\mathrm{PSh}(\mathbf{Alg}_R)$ spanned by the representable presheaves on $\mathbf{Alg}_R$, i.e. by affine $R$-schemes.

Restriction then gives a functor
$$f^*|_{\mathbf{Aff}_R}\colon\mathbf{Aff}_R\longrightarrow\mathbf{Aff}_S$$
sending an affine $R$-scheme $\mathrm{Spec}(A)\rightarrow\mathrm{Spec}(R)$ to the affine $S$-scheme $\mathrm{Spec}(A)\rightarrow\mathrm{Spec}(R)\rightarrow\mathrm{Spec}(S)$.

The fact that $f^*$ sends schemes to schemes (rather than just presheaves on $\mathbf{Alg}_S$) leads us to the following questions:

> **2 Questions.**

 1. Is the image of the restriction of the functors $f_!$ and $f_*$ to $\mathbf{Aff}_R$ contained in the category $\mathbf{Aff}_S$? That is, do $f_!$ and $f_*$ send $R$-schemes to $S$-schemes?
 2. If yes, do they admit a nice description in terms of standard scheme-theoretic constructions?
 3. Do we also get a corresponding adjoint triple of functors between $\mathbf{Aff}_R$ and $\mathbf{Aff}_S$ from the triple $(f_!\dashv f^*\dashv f_*)$?
 4. (Assuming this procedure works), can we globalise it to non-affine schemes?

  [1]: https://i.sstatic.net/405KE.png