Interpret $\Box\Box A$ in the possible worlds

Question

The possible world semantics (Kripke semantics) defines $\Box A$ as follows:
$$
v(\Box A, \omega)=T\iff \forall \omega'\in W\:(\omega R\omega'\land v(A,\omega')=T)
$$
And so
$$
v(\Box\Box A, \omega)=T\iff \forall \omega'\in W\:\forall \omega''\in W\:(\omega R\omega'\land\omega' R\omega''\land v(A,\omega'')=T)
$$
This is clear for temporal logic, i.e. $\Box$ means always. But if $\Box$ means necessary, it is unclear to me. So how is it interpreted for necessary? Any thought is welcome.

Answer 1

The first line says "$A$ necessarily holds in wolrd $w$ iff $A$ holds in all worlds reachable from $w$". You can think of it as, $A$ necessarily holds in wolrd $w$ iff $A$ holds all around as far as one can see from $w$. If you've traveled everywhere in your reach and in all that map you see $A$, you believe that $A$ holds necessarily, it is a law for you.

The second line says "$A$ is necessarily necessary in $w$ iff $A$ holds in all worlds that are reachable from all worlds that are reachable from $w$".

Assume that $R$ is reflexive. If $\Box\Box A$, then by setting $w'=w$ in the interpretation, you get a result equivalent to the interpretation of $\Box A$. This justifies $\Box\Box A \to \Box A$.

Assume that $R$ is transitive. Quoting Alex Kruckman from the comments,

If $A$ is true at every world reachable from $w$, then $A$ is true at every world reachable from every world reachable from $w$, since all such worlds are reachable from $w$.

This justifies $\Box A\to\Box\Box A$. After all, if something is necessary, it is necessarily necessary.