  We present a reconstruction of finite-dimensional quantum theory where all of
the postulates are stated in diagrammatic terms, making them intuitive.
Equivalently, they are stated in category-theoretic terms, making them
mathematically appealing. Again equivalently, they are stated in
process-theoretic terms, establishing that the conceptual backbone of quantum
theory concerns the manner in which systems and processes compose.
  Aside from the diagrammatic form, the key novel aspect of this reconstruction
is the introduction of a new postulate, symmetric purification. Unlike the
ordinary purification postulate, symmetric purification applies equally well to
classical theory as well as quantum theory. Therefore we first reconstruct the
full process theoretic description of quantum theory, consisting of composite
classical-quantum systems and their interactions, before restricting ourselves
to just the `fully quantum' systems as the final step. We propose two novel
alternative manners of doing so, `no-leaking' (roughly that information gain
causes disturbance) and `purity of cups' (roughly the existence of entangled
states). Interestingly, these turn out to be equivalent in any process theory
with cups & caps. Additionally, we show how the standard purification postulate
can be seen as an immediate consequence of the symmetric purification postulate
and purity of cups.
  Other tangential results concern the specific frameworks of generalised
probabilistic theories (GPTs) and process theories (a.k.a. CQM). Firstly, we
provide a diagrammatic presentation of GPTs, which, henceforth, can be subsumed
under process theories. Secondly, we argue that the `sharp dagger' is indeed
the right choice of a dagger structure as this sharpness is vital to the
reconstruction.
