I don't care about using the best monsters. I'm going to use a luminous puma since he looks neat. But what are the sorts of abilities and moves I need on a SYN? And what sort of components should I be feeding it? I'm thinking magic.
Definitely Mana materials for the most part. You want want to check a monster growth spreadsheet.
There's a create your own build section at the end of the Monster Infusion FAQ that discusses noteworthy passives for SYNs. There's another section before it that lists the ways to obtain the passives.
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The way I see it 2 different SYN builds are good to have.
Testudo for quick buffs in random fights, he learns brave faith and GA variants, is early peaker. All he will cast is brave and faith. I also gave him auto faith, item collector, item scavenger II, gilfinder II, magic 35% (earlier just magic 20%), hp 25% (only post game), feral speed II. He still has effectively 2 spare slots, used by resist slow 66% and critical vigilance. Will probably use first strike and auto haste for those.
However later will ideally want a syn thats effective for elemental weaknesses so purple chocobo with same offensive buffs but also the EN buffs. Purple chocobo wont seem much faster than testudo even tho 6 ATB because a lot of the ATB slots get wasted on bad casting order and recasts of EN spells but he will be better than testudo in longer fights against enemies with elemental weaknesses and his buffs will last longer also.
My purple chocobo is similiar to whats in the FAQ except I gave him defense maintenance to help on vigilaga (which runs out so fast otherwise), considering replacing resist magic with resist elements and replacing auto bravery with debuff resistance. Maybe also auto tetra with auto vigilance. Although the last one is probably a bad idea as usually vigilaga is cast which extends vigilance anyway.