Finding the adjoint operation of a black-box differential operator

Question

I have a black-box code (automatic differentiation) that computes $Du$, where $D \in \mathbb{R}^{n \times n}$ and $u \in \mathbb{R}^n$. Note that I do not have the matrix $D$ explicitly and neither do I know its structure. Now I want to compute $D^H u$, where $D^H$ is the Hermitian adjoint of $D$. Is it possible to do this without knowledge of $D$?