Proposition: Complex-linear equivalence need not preserve positivity

An equivalence of complex-linear categories can induce an endomorphism-algebra isomorphism that is not positive with respect to a chosen Hilbert adjoint.

Proof. On a one-object complex-linear category with endomorphism algebra $M_2(\mathbb{C})$, take

This is an algebra automorphism and hence a category equivalence. It sends the Hermitian matrix $X$ to

which is not Hermitian. A positive linear map sends Hermitian matrices to Hermitian matrices, so this automorphism is not positive for the fixed Hilbert structure.