With rel next/previous you ask Google to consider all the pages with these tags to be considered as one page
With a canonical you indicate that the page is a duplicate (or variation) of the canonical url.
Your developers are right - if you put the first page as canonical for the subsequent pages your are basically asking Google to ignore the 2nd, 3rd,... pages which is in conflict with rel next/previous.
It is possible to combine both canonical & rel next/previous - but not in the way as your SEO company is suggesting. Example from Google:
The canonical in this case is used to strip the sessionid which could be a cause for duplicate content.
Please not that if your category pages are split over a lot of pages (like 100) - the rel / next previous stops making sense. In that case it's probably better to focus on optimising the first page & put a "noindex/follow" on the subsequent pages.
Dirk