Yā tesaṃ tesaṃ sattānaṃ tamhi tamhi sattanikāye jāti

Dear friends,

I'm wondering about the relationship between sattānaṃ and jāti in the passage Yā tesaṃ tesaṃ sattānaṃ tamhi tamhi sattanikāye jāti 

Bhikkhu Bodhi's translation is "The birth of the various beings into the various orders of beings"

But the grammar doesn't seem to support exactly this translation, though I admit my Pali grammar understanding is not great.

yā jāti is singular of course, and yā seems to suggest a literal singular birth, which wouldn't really make sense if it was "the birth of beings". sattānaṃ comes before sattanikāye so wouldn't it make better sense to relate the two and translate as something like "that birth in this or that being-group of these or those beings"?

I'm not actually trying to criticize the translation, as it seems fine for general purposes, I'm just trying to understand whether sattānaṃ is possessive of sattanikāye or jāti.

Thanks,

Yuttadhammo