In this note I read, the cofinal map is defined as,
Let $\mathbb D´$ and $\mathbb D$ be two directed sets, $f: \mathbb D´ \to \mathbb D$ is a cofinal map, if $M$ is cofinal in $\mathbb D´$, then $f[M]$ is cofinal in $\mathbb D$.
What´s the difference between the aformentioned definition with merely defining as $f[\mathbb D´]$ is cofinal in $\mathbb D$?
I think they´re supposed to be different, but it seems to me the second, the weaker one, implies the first.
The two definitions are not equivalent. E.g., take $\mathbb{D}' = \mathbb{D} = \mathbb{N}$ with the usual ordering and define $f(n)$ to be $n$ if $n$ is even and to be $0$ if $n$ is odd. Then $f[\mathbb{D}]$ comprises the even numbers and is cofinal (so $f$ satisfies your proposed simpler definition). However, the set $O$ of odd numbers is cofinal, but $f[O] = \{0\}$ is not (so $f$ does not satisfy the definition you quoted from the note).