I want to show that unrestricted grammar is closed under intersection and I don't want to use Turing machine or etc. So I think that we have two grammar $G_1$ and $G_2$ that are restricted for example simple grammars $$ S \rightarrow XY \\ X \rightarrow aXb | ab \\ Y \rightarrow Ya | a$$ and also $G_2$ is $$ S \rightarrow YX \\ X \rightarrow bXa | ba \\ Y \rightarrow Ya | a$$ (I know these are CF grammars but intersection of them is not CF so I was trying to create unrestricted grammar for $G_1 \cap G_2$) so I change these grammars like below to generate $G'$ that is an unrestricted grammar for $G_1 \cap G_2$. $$ S \rightarrow X'Y' \\ X' \rightarrow AX'B | AB \\ Y' \rightarrow Y'A | A \\ BXA \rightarrow X \\ BA \rightarrow X \\ A \rightarrow Y \\ YA \rightarrow Y\\ YX \rightarrow S'$$ but there is two problem that I will forget string that will be accepted because finally the only thing will remain is $S'$ and another problem is that if I have $\lambda$ in the right side of grammar I can't move it to the left side for unrestricted grammar. is my approach true for solving this problem? and what is the idea of solving this?
Edit (complete some part it)
I think after accepting by grammar $G_1$ we can create copy of accepted string by following steps, imagine $aabba$ is accepted by $G_1$ it means that I have $AABBA$ now I can add following rules for every $t \in Terminals$ $$ T \rightarrow T'T'' $$ so $AABBA$ will be $A'A''A'A''B'B''B'B''A'A''$ and now I can define following rules for every $t,r \in Terminals$ $$ T''R' \rightarrow R'T'' $$ now every variable that is the mirror of a terminal with single quotation will jump over variables with double quotation and the result will be $A'A'B'B'A'A''A''B''B''A''$ now I can work with variables with single quotation and after they disappear add rules to convert double quotation variables to terminals, for every $t \in Terminals$ $$ T'' \rightarrow t $$ now just needed to proof that this will not accept string that is not in $L(G_1) \cap L(G_2)$ that I thinks its not complicated and the other thing is that I still don't know what should I do with $\lambda$ in right side of grammar rules.