Let $X,Y$ projective schemes over $k$. Consider the contravariant functor $$\operatorname{Isom}_k(X,Y): \left(\operatorname{Sch}/k\right) \to (\operatorname{Set})$$ $$ S\mapsto \operatorname{Isom}_S(S\times_k X, S\times_k Y) $$ where $\operatorname{Isom}_S(S\times_k X, S\times_k Y)$ is the set of isomorphisms between $S\times_k X$ and $S\times_k Y$ over $S$. I want to prove that this functor is representable.
Recall the following theorem of Grothendieck:
Theorem: Let $X,Y$ projective schemes over $k$. The contravariant functor $$\operatorname{Hom}_k(X,Y): \left(\operatorname{Sch}/k\right) \to (\operatorname{Set})$$ $$ S\mapsto \operatorname{Hom}_S(S\times_k X, S\times_k Y) $$ where $\operatorname{Hom}_S(S\times_k X, S\times_k Y)$ is the set of homomorphisms between $S\times_k X$ and $S\times_k Y$ over $S$, is representable by a disjoint union of quasi projective schemes.
Now the idea is to embeed $\operatorname{Isom}_k(X,Y)$ into $\operatorname{Hom}_k(X,Y)\times \operatorname{Hom}_k(X,Y)$ via the map $f\mapsto (f,g)$ with $fg=id$, $gf=id$. Using the theorem of Grothendieck we know that $\operatorname{Hom}_k(X,Y)\times \operatorname{Hom}_k(X,Y)$ is represented by a scheme $H\times H$, but I'm stuck here. I think that there is a way to a find subscheme of $H\times H$ that represents $\operatorname{Isom}_k(X,Y)$, but I don't know how.
Any hint or ideas? Thank you in advance
Let $ H_{XY} $ be the $ k $-scheme (unique upto a unique isomorphism) representing the functor $ \text{Hom}_k(X,Y) $ according to Grothendieck and let $ H_{YX}, H_{XX} $ and $ H_{YY} $ be defined likewise. There is a map $$ \phi : H_{XY} \times_k H_{YX} \rightarrow H_{XX} \times_k H_{YY} $$ given by specifying it on $ S $-valued points as the obvious map $ (f,g) \rightarrow (gf,fg) $. The scheme $ H_{XX} \times_k H_{YY} $ has a $ k $-valued point $$ e : \operatorname{Spec} k \rightarrow H_{XX} \times_k H_{YY} $$ given by $ (1_X, 1_Y) $, the identity morphisms of the $ k $-schemes $ X,Y$ respectively. Considering the fiber product $ Z $ to be the base change of $ \phi $ wrt $ e $, we easily see that $ Z $ is the desired scheme representing the Isom functor: An $ S $-valued point of $ Z $ is precisely a pair $ (f,g) $ mapping to $ (1_X, 1_Y) $, so is in one-to-one correspondence with the set $ \text{Isom}_k(X,Y)(S) $, via the first coordinate.