How do yarn weight numbers map onto hook sizes?

The 0-7 scale is based on wraps per inch, not on a diameter. #1 super fine is 14-30 wraps per inch, #4 worsted is 9-11, #6 super bulky is 5-6. Wraps per inch is a range wide enough that a thick #4 and a thin #5 can be the same yarn.

For crochet, add roughly 0.5-1 mm to the needle size on the label. A #4 that recommends 4.5 mm needles wants about 4.0-4.5 mm for a firm amigurumi fabric and 5.0-5.5 mm for a drapey one.

A more reliable comparison is metres per 100 g. Lace is 600-900 m, fingering 360-450 m, DK 250-300 m, worsted 180-220 m, bulky 120-150 m. Two yarns with similar metreage per 100 g behave similarly under the hook regardless of what the label says.

This matters most when substituting: a pattern written for a 200 m / 100 g worsted will need about the same total length in any other 200 m / 100 g yarn, even if the fibre is completely different. Buy by length, not by weight.

If you stay within half a category and match metreage, the pattern's stitch counts usually still work. Going up a full category means the same stitch count produces a piece about 20% larger, so either accept a bigger object or drop one increase round.

The one substitution that never works quietly is cotton for acrylic. Cotton has almost no stretch, so a pattern that relies on the fabric relaxing into shape will come out tight and stiff with the same hook and the same count.