Let $X, Y$ be projective irreducible varieties over $k=\mathbb{C}$ and $\varphi: X \dashrightarrow Y$ a rational map. Let $U \subset X$ the maximal open dense subset $U \subset X$ of points where $\varphi$ is regular. The graph $\Gamma_{\varphi}$ is the Zariski closure of the set $\Gamma_U:= \{(x,\varphi(x)) \ \vert \ x \in U \}$ inside $X \times Y$. Obviously $\Gamma_{\varphi}$ is irreducible since $\Gamma_U$ is.
Let now $Z $ be another projective irreducible variety together with a surjective proper map $p: Z \to X$. Define a new rational map $\overline{\varphi}: Z \dashrightarrow Y$ as composition $\varphi \circ p$. Obviously the maximal open dense subset $\overline{U} \subset Z$ where $\overline{\varphi}$ is regular is the preimage $p^{-1}(U)$. Consider now the graph $\Gamma_{\overline{\varphi}}$, the Zariski closure of $\Gamma_{\overline{U}}:= \{ (z,\overline{\varphi}(z) \ \vert \ z \in \overline{U} \}$ inside $Z \times Y$.
Let $p \times \text{id}_Y: Z \times Y \to X \times Y$. This map is still proper. Then $\Gamma_{\overline{U}}=(p \times \text{id})^{-1}(\Gamma_{U})$ and $\Gamma_{\overline{\varphi}} \subset (p \times \text{id})^{-1}(\Gamma_{\varphi})$.
Question: Is the latter inclusion even an equality?
Now comes a bit lengthy background of this question. Since I hope the posed problem can be treated separately, I presume that the following might be less relevaent for this question and could basically be ignored. But for sake of completeness I would like to mention it if someone is interested.
The question is motivated by the problem occuring as intermediate step in this discussion. It is assumed that $X \subset \mathbb{P}^n$ irreducible nondegenerated variety of degree $\ge 2$ (this is just to exclude the case $X$ linear subspace) and assume that general hyperplane section $H \cap X$ spans a $k$-plane $\langle H \cap X \rangle = \Lambda \in \mathbb{G}(k,n)$. This gives rational map
$$\varphi: (\mathbb{P}^{n})^* \dashrightarrow \mathbb{G}(k,n)$$
Assume as before that $U \subset (\mathbb{P}^{n})^*$ is the maximal open dense subset where $\varphi$ is regular, $\Gamma_U := \{(H, \phi(H)) \ \vert \ H \in U \}$ as before and $\Gamma_{\varphi}$ the graph of $\varphi$, i.e. the closure of $\Gamma_U $ in $(\mathbb{P}^{n})^* \times \mathbb{G}(k,n) $.
The crucial step is to show that for every $\Lambda \in \varphi(H):=\Gamma_{\varphi} \cap (\{H\} \times \mathbb{G}(k,n))$ (that is, any point in the image of the fiber of the graph $\Gamma_{\varphi}$, over $H$) the hyperplane section $H \cap X$ is contained in the $k$-plane $\Lambda$.
The textbook suggests that to show it one can apply the irreducibility of the so called universal hyperplane section $\Omega_X:= \{(p, H) \ \vert \ p \in H \cap X \} \subset X \times (\mathbb{P}^{n})^*$ (a Fact which we assume now as blackbox)
My idea here was consider a "new" rational map "lying over" $\varphi$ given by
$$ \overline{\varphi}: \Omega_X \dashrightarrow \mathbb{G}(k,n), \ \ \ (x,H) \mapsto \varphi \circ p(x,H)=\varphi(H) $$
where $p:\Omega_X \to (\mathbb{P}^{n})^*, (x,H) \mapsto H$ is the canonical projection map. We form analogously to $\varphi$ the graph $\Gamma_{\overline{\varphi}}$ as closure of $\Gamma_{\overline{U}} \subset \Omega_X \times \mathbb{G}(k,n)$ where $\overline{U}=p^{-1}(U)$.
Obviously as before $\Gamma_{\varphi}$ and $\Gamma_{\overline{\varphi}}$ are irreducible and it holds $\Gamma_{\overline{U}}=(p \times \text{id})^{-1}(\Gamma_{U})$ and $\Gamma_{\overline{\varphi}} \subset (p \times \text{id})^{-1}(\Gamma_{\varphi})$. As above the crucial question is if the latter inclusion is an equality. Since $\Gamma_{\overline{\varphi}} $ and $(p \times \text{id})^{-1}(\Gamma_{\varphi})$ are of same dimension, this is equivalent to the question if $(p \times \text{id})^{-1}(\Gamma_{\varphi})$ is irreducible.
How does it help to answer affirmatively the question if hyperplane section $H \cap X$ is contained in the $k$-plane $\Lambda \in \Gamma_{\varphi} \cap (\{H\} \times \mathbb{G}(k,n))$? Well, assume we have equality $\Gamma_{\overline{\varphi}} =(p \times \text{id})^{-1}(\Gamma_{\varphi})$.
Intersect $\Gamma_{\overline{\varphi}} \subset \Omega_X \times \mathbb{G}(k,n)$ with closed subset $p_{13}^{-1}(\{ (p,\Lambda) \ \vert \ p \in \Lambda \})$ where $p_{13}: X \times (\mathbb{P}^{n})^* \times \mathbb{G}(k,n) \to X \times \mathbb{G}(k,n)$ is the canonical projection map. The intersection $\Gamma_{\overline{\varphi}} \cap p_{13}^{-1}(\{ (p,\Lambda) \ \vert \ p \in \Lambda \})$ is closed and dense in $\Gamma_{\overline{\varphi}}$ (since it contains the dense subset $\Gamma_{\overline{U}}$ and $\Gamma_{\overline{\varphi}}$ is irreducible. Therefore $p_{13}^{-1}(\{ (p,\Lambda) \ \vert \ p \in \Lambda \})$ contains $\Gamma_{\overline{\varphi}}$. Therefore, since we assumed $\Gamma_{\overline{\varphi}} =(p \times \text{id})^{-1}(\Gamma_{\varphi})$, this means that for every $\Lambda \in \Gamma_{\varphi} \cap (\{H\} \times \mathbb{G}(k,n))$ we have $X \cap H \subset \Lambda$.
And therefore the crucial step becomes to show that $\Gamma_{\overline{\varphi}} \subset (p \times \text{id})^{-1}(\Gamma_{\varphi})$ is an equality.