TITLE: When is push foward/pull back of representable functor representable?
QUESTION [2 upvotes]: Let $\pi : X' \rightarrow X$ be a morphism of schemes (or sites).

For $X$ scheme $T$, what is the pull back $\pi^*h_T$ of the representable functor $h_T$? Is it the fiber product $T\times_{X} X'$?
For $X'$ scheme $T'$, what is the push foward $\pi_*h_{T'}$? When is it representable?

REPLY [1 votes]: 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.
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