Let $H\_1, H\_2$ be two Hilbert schemes parametrizing subschemes in $\mathbb{P}^{n\_1}, \mathbb{P}^{n\_2}$ with Hilbert polynomials $P\_1, P\_2$, respectively. Given a pair $(Z\_1, Z\_2)$ of subschemes in $\mathbb{P}^{n\_1}, \mathbb{P}^{n\_2}$, respectively, we say that $Z\_1 \subset Z\_2$ if there exists a closed immersion $i$ of $\mathbb{P}^{n\_1}$ into $\mathbb{P}^{n\_2}$ such that $i(Z\_1)$ is a closed subscheme of $Z\_2$. Denote by $\mathcal{D}$ the subet of $H\_1 \times H\_2$ consisting of all pairs $(Z\_1, Z\_2)$ such that $Z\_i \in H\_i$ and $Z\_1 \subset Z\_2$. Can we look at it as a scheme?



A question on nested Hilbert scheme
I guess you are looking for a construction that gives an object over $H\_2$, that parametrizes closed immersions $Z\_1 \to Z\_2$ for each $Z\_2 \in H\_2$. If I'm not mistaken, it is the closed subscheme of $Quot(\mathscr{O}\_{\mathbb{P}^{n\_2}}) \times Quot(\mathscr{O}\_{\mathbb{P}^{n\_2}})$ (which parametrizes pairs of ideals) defined by containment of one ideal in the other. Let's call this subscheme $H\_3$. It is a special case of a generalized Hilbert scheme construction whose name I don't know, for any commutative diagram of closed immersions in a projective scheme.


Let $H\_4 = \underline{\operatorname{Imm}}(\mathbb{P}^{n\_1}/\mathbb{P}^{n\_2})$ denote the scheme of closed immersions $\mathbb{P}^{n\_1} \to \mathbb{P}^{n\_2}$ (see page 268 of Grothendieck's [Bourbaki 221](http://www.numdam.org/numdam-bin/item?id=SB_1960-1961__6__249_0) - this is an open subscheme of the Hom scheme). Then we can make a subscheme $X$ of $H\_1 \times H\_4 \times H\_3$ defined by commutativity of the compositions $Z\_1 \to \mathbb{P}^{n\_1} \to \mathbb{P}^{n\_2}$ and $Z\_1 \to Z\_2 \to \mathbb{P}^{n\_2}$.


It seems that the object you want is the image of $X$ under the natural map $H\_1 \times H\_4 \times H\_3 \to H\_1 \times H\_2$ taking $(i\_1, i\_4, Z\_1 \subset Z\_2 \subset \mathbb{P}^{n\_2})$ to $(i\_1, Z\_2 \subset \mathbb{P}^{n\_2})$.


