Why does the proof of undecidability of $A_{TM}$ require the universal TM to take input $\langle M,\langle M\rangle\rangle$? I've read a proof explaining why $A_{\mathrm{TM}}$ is undecidable, and I don't seem to understand why we need to give the opposite of $H$ function $D$ itself as input.

Here's the copy-paste of that proof (link: https://courses.cs.washington.edu/courses/cse322/04au/Lect10.pdf):

$A_{\mathrm{TM}} = \{\langle M,w\rangle\mid M\text{ is a TM that accepts }w\}$ is
undecidable! Proof (by contradiction):


Assume $A_{\mathrm{TM}}$ is decidable  there’s a decider $H$, $L(H)$ = $A_{\mathrm{TM}}$.
$H$ on $\langle M,w\rangle = \mathrm{ACC}$ if $M$ accepts $w$
$\mathrm{REJ}$ if $M$ rejects $w$ (halts in $q_{\mathrm{REJ}}$
or loops on $w$).
Construct new TM $D$: On input $\langle M\rangle$:
Simulate $H$ on $\langle M,\langle M\rangle\rangle$ (here, $w = \langle M\rangle$).
If $H$ accepts, then reject input $\langle M\rangle$;
If $H$ rejects, then accept input $\langle M\rangle$.
What happens when $D$ gets $\langle D\rangle$ as input?
$D$ rejects $\langle D\rangle$ if $H$ accepts $\langle D,\langle D\rangle\rangle$ if $D$ accepts $\langle D\rangle$;
$D$ accepts $\langle D\rangle$ if $H$ rejects $\langle D,\langle D\rangle\rangle$ if $D$ rejects $\langle D\rangle$.
Either way: Contradiction! $D$ cannot exist, so $H$ cannot exist.
Therefore, $A_{\mathrm{TM}}$ is not a decidable language.


So, I don't understand why we need to give $H$ the input of $\langle D,\langle D\rangle\rangle$, why not give it the input of $\langle D,w\rangle$? And give $D$ the same input, isn't that the same thing? Or what's the logic behind the second $\langle D\rangle$? For some reason it's very confusing for me. Thanks.
Why not give it the input of $\langle D,w\rangle$?


What's $w$? The machine $D$ is given some input, which it interprets as the coding of a Turing machine $M$. It's not given any $w$: only $\langle M\rangle$.

Imagine instead that you're given a function $f$ which is defined as $f(x)=x^x$. I ask you to calculate $f(10)$ and your question is a bit like asking, "Why does it have to be $10^{10}$?  Why can't it be $10^y$?"