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


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*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?
 A: 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.
A: In any category $\mathcal{C}$ the Yoneda immersion $Y: \mathcal{C}\to \mathcal{C}^>: x\mapsto h_x$  (some notation of SGA:Seminar of Algebraic Geometry ..by Grothendieck ecc) preserve (and lift) limits (then Pullback too), then the firs assertion if true. ABout pushout isnt true in general categories.
In the following I'm no entirely sure (I have no literature, only an attempt to demonstrate based on the thoughts of the moment)
ABout pushout, I think that is true  for affine schemas and sum (i.e. pushout on $Sch(\mathbb{Z})$) this means that  the contravariant Yoneda immersion for commutative rings $Y: Ring^{op}\to Ring^<: R\mapsto h^R$  preserve sums i.e  
$Ring(R\times S, T)\cong Ring(R, T)\coprod Ring(S, T)$  infact:
if $Sch(T), Sch(R), Sch(S)$ are  connected, then $Sch(R\times S)=Sch(R)\coprod  Sch(S)$) has (only) two connected components, then (let $S$ the schemas category):
$Ring(R\times S, T)\cong S(Sch(T), Sch(R)\coprod  Sch(S))\cong$ 
$S(Sch(T), Sch(R))\coprod  S(Sch(T),Sch(S))\cong Ring(R, T)\coprod Ring (S, T)$
by a decomposition of schemas on connected components (that are associated to othogonals idempotents then these make a decompositions on prodocts of the ring) follow the assertion.
