In Hiragana 4 I think the symbol for "du" is being referred to as "zu", in audio and in text.
And later on zu is still used to refer to its correct Hiragana. Am I wrong? I think it was correct yesterday.
As floomo has said, both ず and づ are "zu"
There is no actual "du" sound in Japanese; the kanji づ IS typed "du" on the keyboard though. It was once pronounced "dzu" as the voiced form of "tsu", but over time it has softened to "zu" and has mostly fallen out of use.
Most cases that originally used this kana have had their spelling changed to use ず "zu" instead, and the old づ you will mostly only see in words with repeating sounds and rendaku, where a consonant that is not usually voiced becomes voiced depending on the sound that comes before it.
Such as つづく - tsu(d)zuku - to continue
A very similar story to the rarely used kana ぢ "ji" (dji / di ) being replaced with じ "ji"
When the sound for "du" is needed in foreign words it is usually written in katakana as either combination Do+small u ドゥ or De+small u デゥ
Do+u is more common, but sometimes you'll see De + small yu デュ "Dyu", a slightly more french-sounding "du" such as in the anime/manga title デュラララ - Durarara
Short answer: ず and づ are pronounced the same, so they are both transcribed as "zu".