Composition of canonical momentum operators

For arbitrary types $C,X$, every family $D^x_{ab}$ of complex-linear endomorphisms of the space of all functions $C→ℂ$, every real number $ℏ$, every $x∈X$, all indices $a,b,c,d∈\{0,1,2\}$, every function $ψ:C→ℂ$, and every $q∈C$, the asserted equality is $[P^x_{ab}(P^x_{cd}ψ)](q)=-ℏ^2[D^x_{ab}(D^x_{cd}ψ)](q)$ in $ℂ$, where $P^x_{uv}$ is defined to be the endomorphism $(-iℏ)D^x_{uv}$ and $ℏ$ on the right is embedded in $ℂ$ before squaring. The inner operator has indices $c,d$ and the outer operator has indices $a,b$. There is no positivity or nonzero assumption on $ℏ$, so the assertion includes $ℏ=0$ and negative $ℏ$; it assumes neither commutation among the operators nor any analytic interpretation as derivatives. No geometry or matter operator is involved. The types may be empty, in which case choices of $q$ or $x$ do not exist and the corresponding quantified claim is vacuous.