Confusion about adjoint of $\nabla$

Question

Let $\Omega$ be an open set in $\mathbb{R}^n$. Consider $\nabla: H^1(\Omega) \to L^2(\Omega)^n$. It is a bounded linear operator. Consider its Hilbert adjoint $\nabla^*: L^2(\Omega)^n \to H^1(\Omega)$, i.e. for $V \in L^2(\Omega)^n$ $$ \langle \phi, \nabla^* V \rangle_{H^1(\Omega)} := \langle \nabla \phi, V \rangle_{L^2(\Omega)^n} \quad \forall\phi\in H^1(\Omega) $$ What is $\nabla^*$? It is not $-\mathrm{div}$, which would be the case if we were considering $\nabla: H^1(\Omega) \subset L^2(\Omega) \to L^2(\Omega)^n$ as an unbounded operator.