diff --git a/src/sage/matroids/basis_exchange_matroid.pyx b/src/sage/matroids/basis_exchange_matroid.pyx
index 016dd3052dc..1fd40569b63 100644
--- a/src/sage/matroids/basis_exchange_matroid.pyx
+++ b/src/sage/matroids/basis_exchange_matroid.pyx
@@ -1863,7 +1863,7 @@ cdef class BasisExchangeMatroid(Matroid):
 
             sage: M = matroids.catalog.N1()
             sage: M._characteristic_setsystem()
-            Iterator over a system of subsets
+            SetSystem of 23 sets over 10 elements
             sage: len(M._characteristic_setsystem())
             23
         """
diff --git a/src/sage/matroids/circuits_matroid.pyx b/src/sage/matroids/circuits_matroid.pyx
index fdc9f20aea0..1b7c5673f92 100644
--- a/src/sage/matroids/circuits_matroid.pyx
+++ b/src/sage/matroids/circuits_matroid.pyx
@@ -543,7 +543,7 @@ cdef class CircuitsMatroid(Matroid):
             sage: from sage.matroids.circuits_matroid import CircuitsMatroid
             sage: M = CircuitsMatroid(matroids.Uniform(2, 4))
             sage: M.circuits()
-            Iterator over a system of subsets
+            SetSystem of 4 sets over 4 elements
             sage: list(M.circuits(0))
             []
             sage: sorted(M.circuits(3), key=str)
@@ -605,7 +605,7 @@ cdef class CircuitsMatroid(Matroid):
             sage: from sage.matroids.circuits_matroid import CircuitsMatroid
             sage: M = CircuitsMatroid(matroids.Uniform(2, 4))
             sage: M.nonspanning_circuits()
-            Iterator over a system of subsets
+            SetSystem of 0 sets over 4 elements
         """
         cdef set NSC = set()
         cdef int i
diff --git a/src/sage/matroids/set_system.pyx b/src/sage/matroids/set_system.pyx
index 5723be38cfd..9ffa2c8e677 100644
--- a/src/sage/matroids/set_system.pyx
+++ b/src/sage/matroids/set_system.pyx
@@ -12,10 +12,8 @@ isomorphism testing.
 AUTHORS:
 
 - Rudi Pendavingh, Stefan van Zwam (2013-04-01): initial version
-
-Methods
-=======
 """
+
 # ****************************************************************************
 #       Copyright (C) 2013 Rudi Pendavingh <rudi.pendavingh@gmail.com>
 #       Copyright (C) 2013 Stefan van Zwam <stefanvanzwam@gmail.com>
@@ -40,7 +38,7 @@ cdef class SetSystem:
 
         sage: M = matroids.catalog.Fano()
         sage: M.circuits()
-        Iterator over a system of subsets
+        SetSystem of 14 sets over 7 elements
 
     To access the sets in this structure, simply iterate over them. The
     simplest way must be::
@@ -75,7 +73,7 @@ cdef class SetSystem:
             sage: from sage.matroids.set_system import SetSystem
             sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]])
             sage: S
-            Iterator over a system of subsets
+            SetSystem of 3 sets over 4 elements
         """
         cdef long i
         if not isinstance(groundset, tuple):
@@ -110,7 +108,7 @@ cdef class SetSystem:
             sage: from sage.matroids.set_system import SetSystem
             sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]])
             sage: S
-            Iterator over a system of subsets
+            SetSystem of 3 sets over 4 elements
             sage: sorted(S[1])
             [3, 4]
             sage: for s in S: print(sorted(s))
@@ -138,7 +136,7 @@ cdef class SetSystem:
             sage: from sage.matroids.set_system import SetSystem
             sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]])
             sage: S
-            Iterator over a system of subsets
+            SetSystem of 3 sets over 4 elements
             sage: len(S)
             3
         """
@@ -196,9 +194,9 @@ cdef class SetSystem:
             sage: from sage.matroids.set_system import SetSystem
             sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]])
             sage: repr(S)  # indirect doctest
-            'Iterator over a system of subsets'
+            'SetSystem of 3 sets over 4 elements'
         """
-        return "Iterator over a system of subsets"
+        return f'SetSystem of {self._len} sets over {self._groundset_size} elements'
 
     cdef copy(self):
         cdef SetSystem S
diff --git a/src/sage/matroids/union_matroid.pyx b/src/sage/matroids/union_matroid.pyx
index ec1be5e70df..c8bd86bf3ae 100644
--- a/src/sage/matroids/union_matroid.pyx
+++ b/src/sage/matroids/union_matroid.pyx
@@ -21,7 +21,7 @@ cdef class MatroidUnion(Matroid):
         {}
         Matroid of rank 1 on 2 elements with 2 bases
         sage: M.bases()
-        Iterator over a system of subsets
+        SetSystem of 2 sets over 5 elements
         sage: list(M.circuits())
         [frozenset({3, 4})]
 
diff --git a/src/sage/matroids/utilities.py b/src/sage/matroids/utilities.py
index 5f376cbcbdd..599b673395c 100644
--- a/src/sage/matroids/utilities.py
+++ b/src/sage/matroids/utilities.py
@@ -70,7 +70,7 @@ def setprint(X):
         sage: from sage.matroids.advanced import setprint
         sage: M = matroids.catalog.Fano().delete('efg')
         sage: M.bases()
-        Iterator over a system of subsets
+        SetSystem of 3 sets over 4 elements
         sage: setprint(M.bases())
         [{'a', 'b', 'c'}, {'a', 'b', 'd'}, {'a', 'c', 'd'}]