It appears that the notion of “left dual” and “right dual” in nLab differs from the literature. If we want “left dual” to coincide with “left adjoint”, shouldn’t we worry about the fact that $\otimes$ is the composition of 1-cells in the *diagrammatic order*? So, $i : I \to A^* \otimes A$ would be $i : I \to AA^*$ in the normal juxtaposition notation. That would mean that $A^*$ is the *left adjoint* of $A$. But, the page dualizable object calls it the “right dual” of $A$.