I have the following problem: let $X$ be a $K$-variety (a geometrically irreducibe $K$-scheme of finite type over, say, a number field $K$). Let $\phi_i:Y_i\rightarrow X$ be a family of dominant morphisms with $Y_i$ $K$-varieties of same dimension of $X$.
Is there a way to construct a morphism $\phi:Y\rightarrow X$ from a $K$-variety $Y$ also of same dimension of $X$ such that the images of the $K$-rational points of the $Y_i$'s are all contained in the image of the ones of $Y$?
Thank you for your suggestions.