Let $C$ be a closed symmetric monidal category. There is hence an adjunction $$ -\otimes X\colon C\leftrightarrows C\colon Map(X,-) $$ involving the internal Hom $Map(-,-)$ for every object $X$ of $C$
An object $X$ of $C$ is called dualizable if the canonical map $$ X\otimes DX\to Map(X,X) $$ is an isomorphism where $DX=Map(X,1)$. It turns out, that this condition is equivalent to the condition that the canonical map $Y\otimes DX\to Map(X,Y)$ is an isomorphism for each $Y$ in $C$. The isomorphism $$ Map(Y,Z\otimes X)\cong Map(Y,Z\otimes DDX)\cong Map(Y,Map(DX,Z))\cong Map(Y\otimes DX, Z) $$ shows that there is an adjunction $$ -\otimes DX\colon C\leftrightarrows C\colon -\otimes X $$ for a dualizable $X$, so then $-\otimes X$ has not only a right adjoint but also a left adjoint.
Is an object $X$ of $C$ necessarily dualizable, if $-\otimes X$ has a left adjoint and does this left adjoint have to be $-\otimes DX$?