EDIT: If the question is for SE level just delete from here as it is also posted there. In fact I have seen some questions in SE regarding the coordinate rings of product of projective varieties but not the coordinate rings of intersection of hypersurfaces in products of projective varieties (maybe it is trivial, but I am not sure)

I am quite new in commutative algebra so I would be grateful if you could clarify to me this issue.

I would like to construct the coordinate ring, $A$, of a Calabi Yau threefold $X$ which is a complete intersection of two hypersurfaces $h_{1}$ and $h_{2}$ in the ambient space $\mathcal{A}=\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{2}$.

We can choose projective coordinates: $x=[x_{0}:x_{1}] $ for $\mathbb{P}^{1}$, $y=[y_{0}:y_{1}:y_{2}]$ for the first $\mathbb{P}^{2}$ and $z=[z_{0}:z_{1}:z_{2}]$ for the other $\mathbb{P}^{2}$. The configuration matrix is: (it is the Schoen manifold)

\begin{equation} X\left[\begin{array}{c||ccc} \mathbb{P}^{1}&1& 1\\ \mathbb{P}^{2}&0& 3\\ \mathbb{P}^{2}&3& 0\\ \end{array}\right]= \begin{cases} x_{0}f_{0}(y)-x_{1}f_{1}(y)=0& \\ x_{0}g_{0}(z)-x_{1}g_{1}(z)=0 & \end{cases} \end{equation} where $f_{0}, f_{1}, g_{0}, g_{1}$ are homogeoneous cubic polynomials.

I know that for a single projective factor, for example the quintic in $\mathbb{P}^{4}$ I can use $A=R_{\mathbb{P}^{4}}/I=\mathbb{C}[x_{0},x_{1},x_{2},x_{3},x_{4}]/\left ( \sum_{i=0}^{4} x_{i}^{5}+\psi x_{0}x_{1}x_{2}x_{3}x_{4} \right )$

However I am unsure about the CICY case. My guess is that I can say that the coordinate ring is the quotient of the graded ring of the ambient space which is the (Segre) embedding $\mathbb{P}^{1}\times \mathbb{P}^{2}\times \mathbb{P}^{2}\hookrightarrow\mathbb{P}^{17}$ by the set of polynomials that vanish on it, i.e.$h_{1}$ and $h_{2}$

\begin{gather} A=R_{\mathbb{P}^{2}\times\mathbb{P}^{2}\times\mathbb{P}^{1}}/{h_{1},h_{2}} =A=R_{\mathbb{P}^{17}}/{h_{1},h_{2}}=(\mathbb{C}[x_{0}y_{0}z_{0},x_{0}y_{0}z_{1},...,x_{1}y_{2}z_{2}]/\left ( x_{0}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{a} y_{0}y_{1}y_{2})-x_{1}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{b} y_{0}y_{1}y_{2}) ),x_{0}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{c} z_{0}z_{1}z_{2})-x_{1}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{d} z_{0}z_{1}z_{2}) \right ) \end{gather}

Or it is just in this way:\begin{gather} A=R/({h_{1},h_{2}}) =(\mathbb{C}[x_{0},x_{1},y_{0},y_{1},y_{2},z_{0},z_{1},z_{2}]/\left ( x_{0}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{a} y_{0}y_{1}y_{2})-x_{1}( \sum_{i=0}^{2} y_{i}^{3}+\psi_{b} y_{0}y_{1}y_{2}) ),x_{0}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{c} z_{0}z_{1}z_{2})-x_{1}( \sum_{i=0}^{2} z_{i}^{3}+\psi_{d} z_{0}z_{1}z_{2}) \right ) \end{gather}

Could you please explain to me the correct reasoning?