$A$ is dually Dedekind infinite (dD-infinite), if there is a surjective non-injective map from $A$ onto $A$.
$A$ is weakly Dedekind infinite (wD-infinite), if there is a surjective map from $A$ onto the natural numbers.
Q1: Why dD-infinite set are wD-infinite set?
Q2: Why TFAE:
(a) $A$ is wD-infinite set
(b) $\mathcal P(A)$ is Dedekind infinite
If $f\colon A\to A$ is surjective and not injective, then it has at least one infinite "orbit". Namely, there is at least one $a\in A$ such that $\{x\in A\mid\exists n\in\Bbb Z, f^n(x)=a\text{ or }f^n(a)=x\}$ is infinite.
Assume otherwise, then each "orbit" is finite and closed under $f$. But for finite sets, surjective implies injective. Therefore on every orbit $f$ is injective, and it must be injective in general.
Fix $a$ whose "orbit" is infinite, and simply consider the function:
$$F(x)=\begin{cases} n & f^n(x)=a\\ 0 & \text{otherwise}\end{cases}$$
In essence, this is repeating the proof that $A$ is Dedekind-infinite if and only if there is an injection from $\Bbb N$ into $A$.