Proving a language is not Turing-recognizable by reduction from $D = \{\langle M\rangle \mid M \text{ rejects input }\langle M\rangle\}$
Like any claim, you have more than one way to prove a language $L$ is unrecognisable.

In case your language is in $Co-RE$ you can do one of the following:


Show $\overline{L}$ is in $RE$ by describing a TM that accepts $\overline{L}$.
Show $\overline{L}$ is in $RE$ by reduction $\overline{L}\le_m Some-Language-In-RE$.
Show $\overline{H_{TM}} \le_m L$ or $\overline{A_{TM}} \le_m L$ or some other language you know is in $Co-RE$


Note that you must show that $L\notin R$ in order to deduce $L\notin RE$.

If $A\notin Co-RE$, you can show a reduction to your language


$A\le_m L$ where $A\notin RE$ so if $L\in RE$ you have a contradiction.


There is more than one way.
I'm not sure I understand your question, but if your task is to show $D\notin RE$.
I would show $\overline{D}\in RE$, and than show $D\notin R$.