I am reading through the construction of $X \times_S Y$, where $X$ and $Y$ are $S$-schemes in Liu's Algebraic Geometry and Arithmetic Curves (Proposition 3.1.2), and I am somewhat stuck justifying a statement that is left unjustified both by Liu and in Hartshorne's book.
Once we have constructed the fiber product in the case where $X, Y$, and $S$ are all affine, and argued that if $U$ is an open subscheme of $X$ and $X \times_S Y$ exists, then so does $U \times_S Y$ and it is isomorphic to $p^{-1}(U)$, where $p$ is the structural morphism from $X \times_S Y \to X$, we want to move to the case where $X$ is not assumed to be affine by taking an open cover $\{X_i\}$ of $X$ by affine schemes and gluing together the $(X_i \times_S Y, p_i, q_i)$. Since each $X_i \times_S Y \cong p_i^{-1}(X_i)$ canonically, we can cover each $X_i \times_S Y$ by the open sets $X_{ij} := p_i^{-1}(X_i \cap X_j)$. We then define isomorphisms $f_{ij} : X_{ij} \to X_{ji}$ by noting that $p_i^{-1}(X_i \cap X_j)$ and $p_j^{-1}(X_i \cap X_j)$ are both canonically isomorphic to $(X_i \cap X_j) \times_S Y$, so we define $f_{ij}$ to be the canonical isomorphism between them.
In order to glue our schemes $p_i^{-1}(X_i)$ together with Liu's Proposition 2.3.33, we need to verify that $f_{ij}(X_{ij} \cap X_{ik}) = X_{ji} \cap X_{jk}$, which amounts to showing that $f_{ij}(p_i^{-1}(X_i \cap X_j \cap X_k)) = p_j^{-1}(X_i \cap X_j \cap X_k)$. Can anyone help me see why this is forced to be true from the definitions?
My attempt has been to use the universal properties - the diagrams are messy, so I'll write it out. Let $\varphi = f_{ij}$ be the canonical isomorphism from $p_i^{-1}(X_i \cap X_j) \to p_j^{-1}(X_i \cap X_j)$, $\hat{\varphi} : p_i^{-1}(X_i \cap X_j \cap X_k) \to p_j^{-1}(X_i \cap X_j \cap X_k)$ be the canonical isomorphism there, $\iota_i : p_i^{-1}(X_i \cap X_j \cap X_k) \to p_i^{-1}(X_i \cap X_j)$, $\iota_j$ defined similarly, and $p_i^Y : p_i^{-1}(X_i \cap X_j) \to Y$ ($p_j^Y$ defined similarly) be the structre map for $p_i^{-1}(X_i \cap X_j) \cong (X_i \cap X_j) \times_S Y$. Then we have that $p_j^Y \circ \iota_j \circ \hat{\varphi} = p_i^Y \circ \iota_i$ since $p_i^Y \circ \iota_i$ and $p_j^Y \circ \iota_j$ are the structure maps to $Y$ for $p_i^{-1}(X_i \cap X_j \cap X_k)$ and $p_j^{-1}(X_i \cap X_j \cap X_k)$, respectively, and $\hat{\varphi}$ is the canonical isomorphism. But then we have that $p_i^Y = p_j^Y \circ \varphi$ for similar reasons, and so we get that $p_i^Y \circ \varphi \circ \iota_i = p_j^Y \circ \iota_j \circ \hat{\varphi}$. But I don't see a reason that $p_i^Y$ would be a monomorphism in general.