Let $E$ be a vector bundle (i.e. locally free $\mathcal{O}_X$-module) on some smooth algebraic variety $X$ and let $\nabla: E \to E \otimes \Omega^1_X$ be an integrable connection.

I have seen that $\nabla$ induces a connection $\nabla^\vee$ on the dual vector bundle $E^\vee=\mathcal{Hom}_{\mathcal{O}_X}(E, \mathcal{O}_X)$.

1) How is $\nabla^\vee$ defined?

2) If $X$ is not complete and $\nabla$ has regular singularities along the boundary, is the same true for $\nabla^\vee$?

3) Are the local systems of horizontal solutions of $\nabla$ and $\nabla^\vee$ dual?