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 $A=R_{\mathbb{P}^{4}}/I=\mathbb{C}[x_{0},x_{1},x_{2},x_{3},x_{4}]/\left ( \sum_{i=0}^{4} x_{i}^{5}+\psi x_{0}x_{1}x_{2}x_{3}x_{4} \right )$

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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}

*Is this correct? Could you please explain to me the correct reasoning?*