  A formal computation proving a new operator identity from known ones is, in
principle, restricted by domains and codomains of linear operators involved,
since not any two operators can be added or composed. Algebraically, identities
can be modelled by noncommutative polynomials and such a formal computation
proves that the polynomial corresponding to the new identity lies in the ideal
generated by the polynomials corresponding to the known identities. In order to
prove an operator identity, however, just proving membership of the polynomial
in the ideal is not enough, since the ring of noncommutative polynomials
ignores domains and codomains. We show that it suffices to additionally verify
compatibility of this polynomial and of the generators of the ideal with the
labelled quiver that encodes which polynomials can be realized as linear
operators. Then, for every consistent representation of such a quiver in a
linear category, there exists a computation in the category that proves the
corresponding instance of the identity. Moreover, by assigning the same label
to several edges of the quiver, the algebraic framework developed allows to
model different versions of an operator by the same indeterminate in the
noncommutative polynomials.
