Okay, let’s proceed step-by-step.

**Step 1: Identify the relevant industry for ITALGAS S.P.A.**  
Italgas S.p.A. is primarily a gas distribution network operator in Italy, a regulated utility. Based on the description, it fits into the **Regulated Utilities** methodology.

**Step 2 & 3: Estimate 2021 Adjusted_EBITDA and FFO**  
For regulated utilities, S&P typically uses EBITDA from operating profit + depreciation, with limited adjustments unless specified.  

- **2021 Operating Profit (Profit Loss From Operating Activities)**: EUR 583,231,000  
- **2021 Depreciation, Amortisation & Impairment**: EUR 445,251,000  
- **2021 EBITDA (Operating Profit + Depreciation)**:  
  \( 583,231,000 + 445,251,000 = 1,028,482,000 \) EUR  

No obvious non-recurring items, pension adjustments, or JV proportional EBITDA provided in the data that would materially change this.  
So, **Adjusted_EBITDA 2021 ≈ EUR 1,028,482,000**.

**FFO 2021 = Adjusted_EBITDA – cash_interest – cash_taxes**  
- Cash interest (Finance Costs Paid): EUR 59,783,000  
- Cash taxes (Income Taxes Paid): EUR 206,923,000  
- FFO = \( 1,028,482,000 - 59,783,000 - 206,923,000 = 761,776,000 \) EUR  

**Step 4: Estimate 2021 Adjusted_Debt**  
- Reported total liabilities 2022-01-01 (i.e., end 2021): EUR 8,009,516,000  
- Short-term financial liabilities: EUR 591,188,000  
- Long-term financial liabilities: EUR 5,785,707,000  
- Total reported debt (ST + LT financial debt): \( 591,188,000 + 5,785,707,000 = 6,376,895,000 \)  
- Cash and equivalents: EUR 1,391,763,000  
- Net debt (reported debt – cash) = \( 6,376,895,000 - 1,391,763,000 = 4,985,132,000 \)  

Without explicit lease/pension deficit/hybrid data to adjust, S&P’s base adjusted debt ≈ net debt.  
So, **Adjusted_Debt 2021 ≈ EUR 4,985,132,000**.

**Step 5 & 6: 2021 Ratios**  
- Adjusted Debt/EBITDA = \( 4,985,132,000 / 1,028,482,000 ≈ 4.85x \)  
- FFO/Adjusted Debt = \( 761,776,000 / 4,985,132,000 ≈ 0.1528 \) (15.3%)

**Step 7 & 8: Estimate 2022 Adjusted_EBITDA and FFO**  
- 2022 Operating Profit: EUR 641,338,000  
- 2022 Depreciation: EUR 479,186,000  
- EBITDA = \( 641,338,000 + 479,186,000 = 1,120,524,000 \) EUR  
- Adjusted_EBITDA 2022 ≈ EUR 1,120,524,000  

FFO 2022:  
- Cash interest paid: EUR 53,720,000  
- Cash taxes paid: EUR 152,372,000  
- FFO = \( 1,120,524,000 - 53,720,000 - 152,372,000 = 914,432,000 \) EUR  

**Step 9: Estimate 2022 Adjusted_Debt**  
- ST financial liabilities: EUR 142,437,000  
- LT financial liabilities: EUR 6,402,913,000  
- Total debt: \( 142,437,000 + 6,402,913,000 = 6,545,350,000 \)  
- Cash: EUR 451,946,000  
- Net debt = \( 6,545,350,000 - 451,946,000 = 6,093,404,000 \)  
- Adjusted_Debt 2022 ≈ EUR 6,093,404,000  

**Step 10 & 11: 2022 Ratios**  
- Adjusted Debt/EBITDA = \( 6,093,404,000 / 1,120,524,000 ≈ 5.44x \)  
- FFO/Adjusted Debt = \( 914,432,000 / 6,093,404,000 ≈ 0.1501 \) (15.0%)

**Step 12: Year-on-year trend**  
- Leverage change: \( 5.44x - 4.85x = +0.59x \) → **Deteriorating**  
- FFO/Debt change: \( 0.1501 - 0.1528 = -0.0027 \) → **Stable** (within ±0.03)  

One signal is Stable, the other is Deteriorating → net category is **Deteriorating**.

<answer>Deteriorating</answer>