"The butterfly is partly blue and partly yellow."
Translation:Tá an féileacán idir a bheith gorm agus buí.
Don't try to break it down grammatically. It's an idiomatic phrase. The a is not a possessive adjective, it's preposition, and the closest equivalent to a bheith is the English infinitive "to be", but that's not going to help you analyze this turn of phrase from the point of view of English grammar.
Actually, it does help. Different languages have different logic(s), and when I see the logic I can remember what to do, whereas when it's pure memorization, I find it much, much harder. So knowing the closest equivalent really does help. I suppose I'm an idiosyncratic learner. Anyway, thanks.
That's too literal a translation of "idir" - idir ghorm agus bhuí actually means "both blue and yellow", not "between blue and yellow" - the a bheith just implies that it is being blue and yellow at the same time, so partly blue and partly yellow.
Here's a few snippets from a search for idir a bheith - the meaning changes with the context.
Tá cumasc anois ann idir a bheith ag ithe sa bhaile agus ag ithe amuigh - " There is a merging of eating at home and eating out"
fear a roinníonn a chuid ama idir a bheith ina mhúinteoir tacaíochta agus ina fhoilsitheoir leabhar agus CD-ROM trí Ghaeilge "A man who divides his time between being a support teacher and a publisher of books and CD-ROMs in Irish"
an difríocht idir a bheith fostaithe agus féinfhostaithe - "the difference between being employed and self-employed"