diff --git a/doc/source/overview/tutorials/pdf.ipynb b/doc/source/overview/tutorials/pdf.ipynb
index bfc6ad3ad..38ddcd63d 100644
--- a/doc/source/overview/tutorials/pdf.ipynb
+++ b/doc/source/overview/tutorials/pdf.ipynb
@@ -74,7 +74,7 @@
     "from ekobox.apply import apply_pdf\n",
     "\n",
     "with eko.EKO.read(\"./myeko.tar\") as evolution_operator:\n",
-    "    evolved_pdfs = apply_pdf(evolution_operator, pdf)"
+    "    evolved_pdfs, _integration_errors = apply_pdf(evolution_operator, pdf)"
    ]
   },
   {
@@ -82,7 +82,9 @@
    "id": "0c2e752c-4a7c-4e2a-bd97-94f54907184a",
    "metadata": {},
    "source": [
-    "The returned object `evolved_pdfs` is a dictionary which maps the requested final scales onto further dictionaries:"
+    "The function returns two dictionaries, one filled with PDF values for each final scale (`evolved_pdfs`), \n",
+    "and the other containing the integration errors (`_integration_errors`). \n",
+    "Both of them have the same structure, for example, let's inspect `evolved_pdfs` :"
    ]
   },
   {
@@ -111,8 +113,8 @@
    "id": "0ddf91bb-bc2c-4833-b624-32c5ac2d717a",
    "metadata": {},
    "source": [
-    "Each final scale contains a dictionary with a key `pdfs`, where all PDF values are hold, and a key `errors` , where the integration errors are hold.\n",
-    "Finally, inside each of those there is a mapping of Monte Carlo particle identifiers onto a the PDF value at the requested interpolation points.\n",
+    "Each final scale contains a dictionary, where all PDF values are hold.\n",
+    "Those are mapped with the Monte Carlo particle identifiers onto a the PDF value at the requested interpolation points.\n",
     "\n",
     "E.g. to access the gluon PDF at $Q^2 = 10000\\,\\text{GeV}^2$ you can run:"
    ]
@@ -136,7 +138,7 @@
     }
    ],
    "source": [
-    "evolved_pdfs[(10000.0, 5)][\"pdfs\"][21]"
+    "evolved_pdfs[(10000.0, 5)][21]"
    ]
   },
   {
@@ -222,13 +224,6 @@
    "id": "8c18c327",
    "metadata": {},
    "outputs": [
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "Overwriting old PDF installation\n"
-     ]
-    },
     {
      "name": "stdout",
      "output_type": "stream",
@@ -263,7 +258,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "LHAPDF 6.4.0 loading /home/felix/local/share/LHAPDF/Evolved_PDF/Evolved_PDF_0000.dat\n",
+      "LHAPDF 6.5.0 loading /Users/giacomomagni/.conda/envs/eko_dev/share/LHAPDF/Evolved_PDF/Evolved_PDF_0000.dat\n",
       "Evolved_PDF PDF set, member #0, version 1\n"
      ]
     }
@@ -339,7 +334,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "LHAPDF 6.4.0 loading /home/felix/local/share/LHAPDF/CT14llo/CT14llo_0000.dat\n",
+      "LHAPDF 6.5.0 loading /Users/giacomomagni/.conda/envs/eko_dev/share/LHAPDF/CT14llo/CT14llo_0000.dat\n",
       "CT14llo PDF set, member #0, version 1; LHAPDF ID = 13205\n"
      ]
     }
@@ -384,17 +379,10 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 22,
    "id": "05b81dca",
    "metadata": {},
    "outputs": [
-    {
-     "name": "stderr",
-     "output_type": "stream",
-     "text": [
-      "Overwriting old PDF installation\n"
-     ]
-    },
     {
      "name": "stdout",
      "output_type": "stream",
@@ -424,7 +412,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 23,
    "id": "2b02aab1",
    "metadata": {},
    "outputs": [
@@ -432,34 +420,34 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "LHAPDF 6.4.0 loading /home/felix/local/share/LHAPDF/my_ct14llo/my_ct14llo_0000.dat\n",
-      "my_ct14llo PDF set, member #0, version 1\n",
+      "LHAPDF 6.5.0 loading /Users/giacomomagni/.conda/envs/eko_dev/share/LHAPDF/my_ct14llo/my_ct14llo_0000.dat\n",
       "           x            Q2       ct14llo    my_ct14llo  relative_diff\n",
-      "0   0.000010     25.000000  7.635785e+01  7.630719e+01       0.000663\n",
-      "1   0.000173     25.000000  3.194273e+01  3.192239e+01       0.000637\n",
-      "2   0.003000     25.000000  1.081843e+01  1.081160e+01       0.000632\n",
+      "0   0.000010     25.000000  7.635785e+01  7.630727e+01       0.000662\n",
+      "1   0.000173     25.000000  3.194273e+01  3.192242e+01       0.000636\n",
+      "2   0.003000     25.000000  1.081843e+01  1.081161e+01       0.000631\n",
       "3   0.051962     25.000000  1.958956e+00  1.958820e+00       0.000069\n",
-      "4   0.900000     25.000000  1.922415e-05  1.955440e-05      -0.017179\n",
-      "5   0.000010    111.803399  1.333957e+02  1.333028e+02       0.000697\n",
-      "6   0.000173    111.803399  4.777286e+01  4.773855e+01       0.000718\n",
+      "4   0.900000     25.000000  1.922415e-05  1.955338e-05      -0.017126\n",
+      "5   0.000010    111.803399  1.333957e+02  1.333029e+02       0.000696\n",
+      "6   0.000173    111.803399  4.777286e+01  4.773856e+01       0.000718\n",
       "7   0.003000    111.803399  1.341028e+01  1.340044e+01       0.000734\n",
       "8   0.051962    111.803399  1.978216e+00  1.978292e+00      -0.000038\n",
-      "9   0.900000    111.803399  6.644805e-06  6.756354e-06      -0.016787\n",
-      "10  0.000010    500.000000  1.967032e+02  1.965517e+02       0.000770\n",
-      "11  0.000173    500.000000  6.291393e+01  6.286327e+01       0.000805\n",
+      "9   0.900000    111.803399  6.644805e-06  6.754169e-06      -0.016459\n",
+      "10  0.000010    500.000000  1.967032e+02  1.965518e+02       0.000770\n",
+      "11  0.000173    500.000000  6.291393e+01  6.286326e+01       0.000805\n",
       "12  0.003000    500.000000  1.542347e+01  1.541073e+01       0.000826\n",
       "13  0.051962    500.000000  1.947465e+00  1.947532e+00      -0.000034\n",
-      "14  0.900000    500.000000  2.929060e-06  2.979511e-06      -0.017224\n",
-      "15  0.000010   2236.067977  2.633266e+02  2.631189e+02       0.000789\n",
+      "14  0.900000    500.000000  2.929060e-06  2.977019e-06      -0.016373\n",
+      "15  0.000010   2236.067977  2.633266e+02  2.631190e+02       0.000789\n",
       "16  0.000173   2236.067977  7.708540e+01  7.702204e+01       0.000822\n",
       "17  0.003000   2236.067977  1.700410e+01  1.699004e+01       0.000827\n",
       "18  0.051962   2236.067977  1.893923e+00  1.894094e+00      -0.000090\n",
-      "19  0.900000   2236.067977  1.544450e-06  1.572860e-06      -0.018395\n",
-      "20  0.000010  10000.000000  3.314097e+02  3.311450e+02       0.000799\n",
-      "21  0.000173  10000.000000  9.023010e+01  9.015576e+01       0.000824\n",
-      "22  0.003000  10000.000000  1.825934e+01  1.824477e+01       0.000798\n",
+      "19  0.900000   2236.067977  1.544450e-06  1.570506e-06      -0.016870\n",
+      "20  0.000010  10000.000000  3.314097e+02  3.311451e+02       0.000799\n",
+      "21  0.000173  10000.000000  9.023010e+01  9.015575e+01       0.000824\n",
+      "22  0.003000  10000.000000  1.825934e+01  1.824476e+01       0.000798\n",
       "23  0.051962  10000.000000  1.830992e+00  1.831291e+00      -0.000163\n",
-      "24  0.900000  10000.000000  9.288458e-07  9.463689e-07      -0.018866\n"
+      "24  0.900000  10000.000000  9.288458e-07  9.442889e-07      -0.016626\n",
+      "my_ct14llo PDF set, member #0, version 1\n"
      ]
     }
    ],