When reading papers on the no-cloning theorem, I understand that one is searching for a unitary matrix $U$, such that for any state $\vert \phi_A \rangle$ in a Hilbert state $H_A$, we have that $U\Big(\vert \phi \rangle_A \otimes \vert e \rangle_B\Big) = \vert \phi \rangle_A \otimes \vert \phi \rangle_B$, where $e_B$ is a blank state in the Hilbert space $H_B$, and $\vert \phi \rangle_B$ is a copy of $\vert \phi \rangle_A$ in $H_B$. Such $U$ cannot exist.
Still, I wonder what "blank state'' really means -- is one allowed to choose $\vert e \rangle_B$ as a function of $\vert \phi \rangle_A$ when building cloning machines, or generalizations ? (So that the cloning machine not only comes with the matrix $U$, but also with a choice function.)