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An eidolon has it's summoner's alignment, so you can smite it.
Summoner wrote:
An eidolon has the same alignment as the summoner that calls it and can speak all of his languages.
However Core eidolons don't have alignment subtypes, so the paladin wouldn't get extra damage for smiting an outsider with the evil subtype - unless the eidolon has the Fiendish Appearance evolution.
Paladin wrote:
If the target of smite evil is an outsider with the evil subtype, an evil-aligned dragon, or an undead creature, the bonus to damage on the first successful attack increases to 2 points of damage per level the paladin possesses.
Fiendish Appearance wrote:
The eidolon appears as a fiendish creature and manifests some of the abilities of a fiend. Spells and effects that target creatures with the evil subtype or have specific effects against such creatures affect the eidolon as if it had that subtype. The eidolon gains a +2 bonus on saving throws against acid, disease, fire, and poison spells and effects. It also gains an amount of spell resistance equal to 5 + its HD against spells with the good descriptor. The summoner must be evil-aligned to select this evolution.
The Unchained summoner's eidolon does get subtypes to match its alignment and so would take the extra damage for being an outsider (evil) - though note that an Unchained eidolon's alignment doesn't always match its summoner so an evil summoner could have a neutral eidolon that couldn't be smote at all.
Unchained Summoner wrote:
An eidolon must be within one alignment step of the summoner who calls it (so a neutral good summoner can call a neutral, lawful good, or chaotic good eidolon) and can speak all of his languages.
Unchained Eidolon wrote:
Alignment: The eidolon is a creature of this alignment. If its alignment is chaotic, evil, lawful, or good, it gains that subtype.