Choi-Jamiołkowski isomorphism(s) and channel/state duality, naturally!

Provided a maximally entangled state vector in $\mathcal{H}\otimes \mathcal{H}$, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of $\mathsf{C}P(\mathcal{H},\mathcal{K})$ (completely positive maps of density operators on $\mathcal{H}$ tothose on $\mathcal{K}$) onto the density operators on $\mathcal{K}\otimes \mathcal{H}$, which extends to a unitary map. Here, $\mathcal{H}$ and $\mathcal{K}$ are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map’') natural in both $\mathcal{H}$ and $\mathcal{K}$ which produces density operators on either $\mathsf{L}(\mathcal{H},\mathcal{K})$ (Hilbert-Schmidt inner product) or $\mathcal{K}\otimes {\mathcal{H}}^{\ast }$ instead of $\mathcal{K}\otimes \mathcal{H}$. In particular, no entangled state input is required. There are no such binatural maps carried on $\mathcal{K}\otimes \mathcal{H}$, but those natural in only $\mathcal{K}$ are canonically associated with unitary maps $\mathcal{H}\to {\mathcal{H}}^{\ast }$ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.