Show Fiber Product of Rational Elliptic Surfaces is Calabi-Yau

The diagonal $\Delta $ is linearly equivalent to $\{p\}\times \mathbb{P}^1 +\mathbb{P}^1\times \{p\} $ for any $p$ in $\mathbb{P}^1$. Therefore $X$ is the zero locus in $S\times S'$ of a section of $L:=\pi^*\mathcal{O}(1) \boxtimes \pi'^*\mathcal{O}(1) $. On the other hand, standard theory of elliptic surfaces gives $\omega _S=\pi ^*\mathcal{O}(-1) $ and $\omega _{S'}=\pi' ^*\mathcal{O}(-1) $, therefore $\omega _{S\times S'}= L^{-1}$. Then the adjunction formula gives indeed $\omega_X\cong \mathcal{O}_{X}$.

$S\times_{\mathbb{P}^1}S'$ is a complete intersection in $\mathbb{P}^1\times \mathbb{P}^2\times \mathbb{P}^2$: it is given by two equations of degree $(1,3,0)$ and $(1,0,3)$. The canonical bundle is trivial by the adjunction formula.