We consider the tensorial Schur productR∘ ⊗S= [ rij⊗ sij] for R∈ Mn(A) , S∈ Mn(B) , with A,BunitalC∗-algebras, verify that such a ‘tensorial Schur product’ of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Choi that a linear map ϕ: Mn→ Md is completely positive if and only if [ϕ(Eij)]∈Mn(Md)+, where of course Eij: 1 ≤ i, j≤ n denotes the usual system of matrix units in Mn(: = Mn(C)). We also discuss some other corollaries of the main result. © 2015, Springer Basel.