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Suppose that $P$ is a forcing notion, $L$ is a language (in $V$),$\dot{\tau}$ a $P$-name such that $P\Vdash$ "$\dot{\tau}$ is a countable$L$-structure". In the product $P\times P$, there are names$\dot{\tau_{1}},\dot{\tau_{2}}$ such that for any generic filter $G=G_{1}\timesG_{2}$ over $P\times P$, $\dot{\tau}_{1}[G]=\dot{\tau}[G_{1}]$ and$\dot{\tau}_{2}[G]=\dot{\tau}[G_{2}]$. Zapletal asked whether or not $P \timesP \Vdash \dot{\tau}_{1}\cong\dot{\tau}_{2}$ implies that there is some $M\in V$such that $P \Vdash \dot{\tau}\cong\check{M}$. We answer this negatively anddiscuss related issues.

Forcing a countable structure to belong to the ground model