The drawing convention used in the cartoon guide strongly suggests that you’re making an incorrect substitution when you think about this.
When you draw “PA says (X)”, that corresponds in the symbolic proof of Löb’s theorem to “PA⊢X”, and when you draw “PA says (PA says X)” that corresponds to PA⊢◻X. The notation makes it look like “PA⊢◻X” is just the second order version of “PA⊢X”, i.e. a second application of “PA says” (PA⊢). But of course “PA⊢(PA⊢X)” is the same as “P⊢X”, not PA⊢◻X.
The drawing convention used in the cartoon guide strongly suggests that you’re making an incorrect substitution when you think about this.
When you draw “PA says (X)”, that corresponds in the symbolic proof of Löb’s theorem to “PA⊢X”, and when you draw “PA says (PA says X)” that corresponds to PA⊢◻X. The notation makes it look like “PA⊢◻X” is just the second order version of “PA⊢X”, i.e. a second application of “PA says” (PA⊢). But of course “PA⊢(PA⊢X)” is the same as “P⊢X”, not PA⊢◻X.