NAT translates addresses that their DHCP gives out (and not only them).
ima75 answered on March 19th 20 at 09:17
These are two completely different technologies. They go hand in hand, but no problem to use them separately.
DHCP without NAT — any domoset with "white" addresses, which automatically distributes them.
NAT no DHCP no problem to do it on an ordinary home router.
How does NAT. Take and adjust the outgoing package some of the parameters like outbound port (to find out whose package). At the same time change the outbound address to an external address network. When it comes to the answer, restore the original caller, change the incoming address to the address of that person and retranslates it to the internal network.
The task is simple routing (not NAT) — just pass the packet on down the chain by reducing TTL and defining a routing table in which connection it is necessary to pass it.