Creating an OCF based on any function from ordinals to ordinals

Take any function F from ordinals to ordinals. Then, we define:

• ψ₀^F(0)[0] = 0
• ψ_β+1^F(0) = min{x|"x is regular"∧∀y<min{z|z=ψ_z^F(0)},w<β(ψ_w^F(y),cof(ψ_w^F(y))<x)}
• ψ_β+1^F(0)[n] = n
• cof(β) ≥ ω: ψ_β^F(0)[n] = ψ_β[n]^F(0)
• α+1 > 0: ψ_β^F(α+1)[0] = ψ_β^F(α)+1
• α+1 > 0: ψ_β^F(α+1)[n+1] = F(ψ_β^F(α+1)[n])
• ω ≤ cof(α) < ψ_β+1^F(0): ψ_β^F(α)[n] = ψ_β^F(α[n])
• cof(α) = ψ_β+1^F(0): ψ_β^F(α)[0] = 0
• cof(α) = ψ_β+1^F(0): ψ_β^F(α)[n+1] = ψ_β^F(α[ψ_β^F(α)[n]])
• cof(α) > ψ_β+1^F(0): ψ_β^F(α) = ψ_β^F(ψ_β+1^F(α))

The cofinalities of some ordinals are not taken into account and unspecified, but these are either (1) supposed to already be known by the reader or (2) able to be deduced from the ruleset. (NOTE: There are no extended FSes, like ω²[ω+1] or Ω[Ω₂] (although something like Ω₂[Ω] is fine, as the FS of an ordinal has the same length as its cofinality))