Gotta love the incredible disinformation available on the internet. Yes, 700c is the rim size that a 29" tire will mount. Narrow or wide, any 29" tire will/should fit to any 700c rim.
The "C" used to mean something but doesn't any more. It does not mean clincher or centimeter. It used to be a French sizing system. There were A, B, C, D sizes and the letters corresponded to widths. An "A" would be narrower than a "B" which would be narrower than "C" which would be narrower than "D". Today, 700C really means any rim that has a bead seat diameter of 622mm. There is also a 700D rim size which as a bead seat diameter of 587mm.
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