Let's we have a primal model like
$\max~~ x + \langle I, Z \rangle$
$s.t. ~~~Ax + y I - Z \preceq B$
$~~~~~~~~~Z \succeq 0, ~X \geq 0, ~~y ~free$
where $A, B \in {\mathbb R^{n \times n}}$. The capital letters means they are in the matrix form and small letters means a vector form. Introducing the (matrix) variable $Y$ for the dual, one can find the dual model as
$\min ~~\langle B, Y \rangle$
$s.t.~~~\langle A, Y \rangle \geq 1$
$ ~~~~~~~~~\langle I, Y \rangle = 0 $
$ ~~~~~~~~~-Y \geq 1$
$~~~~~~~~~Y \succeq 0$
The last constraints and the positive semidefiniteness of variable $Y$ does not make sense together. Have I written the dual in the right form and if not what would be the correct one?