Why elemental focus and spell echo can support spectre?

I'm a beginner in this game. I'm trying to understand the summoner build for Raise Spectre - Slave Driver

According to the description of elemental focus, supported skills can deal more elemental damage. However, spectre should be applied as minion damage not elemental damage.

For spell echo, supported skills can deal more spell damage but spectre deals minion damage not spell damage.

https://pathofexile.gamepedia.com/Raise_Spectre

Read support gem section.

Spectres generally have the same abilities as the monsters they are raised from. If the monsters can cast a spell, so can the spectre.

You are right about only minion damage mods in your tree or gear affecting spectre damage, but support gems can be used as outlined in the link above.

Thanks, guys. I get it.

The spectre I raised can deal elemental damage as well, depending on the original ability from that corpse when it is alive.

Bingo!