{\ensuremath{\otimes}}{\ensuremath{\mathbb{C}}}^2$ defined by $${{|} 0\rangle} \mapsto {{|} 00\rangle}, \qquad {{|} 1\rangle} \mapsto {{|} 11\rangle} .$$ By construction, this copies the elements of the computational basis — and *only* these, as in general $$\delta_{\text{std}}(\alpha{{|} 0\rangle}+\beta{{|} 1\rangle}) =
  \alpha{{|} 00\rangle}+\beta{{|} 11\rangle} \neq (\alpha{{|} 0\rangle}+\beta{{|} 1\rangle})
  {\ensuremath{\otimes}}(\alpha{{|} 0\rangle}+\beta{{|} 1\rangle}).$$ Next, consider *deleting maps* $\varepsilon \colon H \to
{\ensuremath{\mathbb{C}}}$ by linearly extending ${{|} e_i\rangle} \mapsto 1$. In contrast to copying, these can be defined in *finite dimension only*. It is straightforward to verify that these maps define a dagger Frobenius structure on $H$. Moreover, the following result provides a striking converse.

\[cpvth\] [@coeckepavlovicvicary:bases] Orthonormal bases of a finite-dimensional Hilbert space $H$ are in one-to-one correspondence with dagger Frobenius structures on $H$.

This result in fact follows easily from previous results in the literature on Frobenius algebras [@abrams:thesis]; we will give a short proof from the established literature in Section \[subsec:furtherconditions\].

Another result provides a counterpart—at first sight displaying very different looking behaviour—in the category ${{\ensuremath{\mathbf{Rel}}}}$.

[@pavlovic:frobeniusinrel] Dagger Frobenius structures in the category ${{\ensuremath{\mathbf{Rel}}}}$ correspond to disjoint unions of abelian groups.

We shall provide a different proof of this result in Section \[frobrelsec\], which makes no use of units, and hence generalizes to a wide 