Let $\pi : X' \rightarrow X$ be a morphism of schemes (or sites).


1. For $X$ scheme $T$, what is the pull back $\pi^\*h\_T$ of the representable functor $h\_T$? Is it the fiber product $T\times\_{X} X'$?
2. For $X'$ scheme $T'$, what is the push foward $\pi\_\*h\_{T'}$? When is it representable?



When is push foward/pull back of representable functor representable?
This question has been answered in the comments. The former is the fibered product and the latter is the Weil restriction. The former always exists, but the latter does not. I am reposting this as a CW answer; if it gets upvoted, this question will not reappear on the front page.


