diff --git a/src/seemps/state/canonical_mps.py b/src/seemps/state/canonical_mps.py
index afe17e6..2848b40 100644
--- a/src/seemps/state/canonical_mps.py
+++ b/src/seemps/state/canonical_mps.py
@@ -7,8 +7,8 @@
 from seemps.state.core import (
     DEFAULT_STRATEGY,
     Strategy,
-    _update_in_canonical_form_right,
-    _update_in_canonical_form_left,
+    _update_canonical_right,
+    _update_canonical_left,
     _update_canonical_2site_left,
     _update_canonical_2site_right,
     _canonicalize,
@@ -244,11 +244,11 @@ def update_canonical(
             The truncation error of this update.
         """
         if direction > 0:
-            self.center, err = _update_in_canonical_form_right(
+            self.center, err = _update_canonical_right(
                 self._data, A, self.center, truncation
             )
         else:
-            self.center, err = _update_in_canonical_form_left(
+            self.center, err = _update_canonical_left(
                 self._data, A, self.center, truncation
             )
         self.update_error(err)
diff --git a/src/seemps/state/core.cc b/src/seemps/state/core.cc
index f4767bd..f933f90 100644
--- a/src/seemps/state/core.cc
+++ b/src/seemps/state/core.cc
@@ -135,14 +135,14 @@ PYBIND11_MODULE(core, m) {
 
   m.def("schmidt_weights", &schmidt_weights);
   m.def(
-      "_update_in_canonical_form_right", &_update_in_canonical_form_right,
-      py::arg("state"), py::arg("tensor"), py::arg("site"), py::arg("strategy"),
+      "_update_canonical_right", &_update_canonical_right, py::arg("state"),
+      py::arg("tensor"), py::arg("site"), py::arg("strategy"),
       py::arg("overwrite") = false,
       R"doc(Insert a tensor in canonical form into the MPS Ψ at the given site.
     Update the neighboring sites in the process)doc");
   m.def(
-      "_update_in_canonical_form_left", &_update_in_canonical_form_left,
-      py::arg("state"), py::arg("tensor"), py::arg("site"), py::arg("strategy"),
+      "_update_canonical_left", &_update_canonical_left, py::arg("state"),
+      py::arg("tensor"), py::arg("site"), py::arg("strategy"),
       py::arg("overwrite") = false,
       R"doc(Insert a tensor in canonical form into the MPS Ψ at the given site.
     Update the neighboring sites in the process)doc");
diff --git a/src/seemps/state/core.pyi b/src/seemps/state/core.pyi
index b8684c9..cf671ca 100644
--- a/src/seemps/state/core.pyi
+++ b/src/seemps/state/core.pyi
@@ -77,10 +77,10 @@ def right_orth_2site(
 ) -> tuple[Tensor3, Tensor3, float]: ...
 def _destructive_svd(A: np.ndarray) -> tuple[np.ndarray, np.ndarray, np.ndarray]: ...
 def schmidt_weights(A: np.ndarray) -> np.ndarray: ...
-def _update_in_canonical_form_left(
+def _update_canonical_left(
     state: list[Tensor3], A: Tensor3, site: int, truncation: Strategy
 ) -> tuple[int, float]: ...
-def _update_in_canonical_form_right(
+def _update_canonical_right(
     state: list[Tensor3], A: Tensor3, site: int, truncation: Strategy
 ) -> tuple[int, float]: ...
 def _update_canonical_2site_left(
diff --git a/src/seemps/state/mps.h b/src/seemps/state/mps.h
index 96748e3..b9831df 100644
--- a/src/seemps/state/mps.h
+++ b/src/seemps/state/mps.h
@@ -17,14 +17,14 @@ Weight scprod(py::object A, py::object B);
 
 py::object schmidt_weights(py::object A);
 
-std::tuple<int, double>
-_update_in_canonical_form_right(py::list state, py::object A, int site,
-                                const Strategy &truncation,
-                                bool overwrite = false);
-std::tuple<int, double>
-_update_in_canonical_form_left(py::list state, py::object A, int site,
-                               const Strategy &truncation,
-                               bool overwrite = false);
+std::tuple<int, double> _update_canonical_right(py::list state, py::object A,
+                                                int site,
+                                                const Strategy &truncation,
+                                                bool overwrite = false);
+std::tuple<int, double> _update_canonical_left(py::list state, py::object A,
+                                               int site,
+                                               const Strategy &truncation,
+                                               bool overwrite = false);
 double _canonicalize(py::list state, int center, const Strategy &truncation);
 std::tuple<py::object, py::object, double>
 left_orth_2site(py::object AA, const Strategy &strategy);
diff --git a/src/seemps/state/schmidt.cc b/src/seemps/state/schmidt.cc
index 7c3e086..3bf8a55 100644
--- a/src/seemps/state/schmidt.cc
+++ b/src/seemps/state/schmidt.cc
@@ -5,12 +5,13 @@
 
 namespace seemps {
 
-std::tuple<int, double>
-_update_in_canonical_form_right(py::list state, py::object A, int site,
-                                const Strategy &strategy, bool overwrite) {
+std::tuple<int, double> _update_canonical_right(py::list state, py::object A,
+                                                int site,
+                                                const Strategy &strategy,
+                                                bool overwrite) {
   if (!is_array(A) || array_ndim(A) != 3) {
     throw std::invalid_argument(
-        "Invalid tensor passed to _update_in_canonical_form_right");
+        "Invalid tensor passed to _update_canonical_right");
   }
   py::object tensor = overwrite ? array_getcontiguous(A) : array_copy(A);
   auto a = array_dim(tensor, 0);
@@ -28,13 +29,13 @@ _update_in_canonical_form_right(py::list state, py::object A, int site,
   return {site, err};
 }
 
-std::tuple<int, double> _update_in_canonical_form_left(py::list state,
-                                                       py::object A, int site,
-                                                       const Strategy &strategy,
-                                                       bool overwrite) {
+std::tuple<int, double> _update_canonical_left(py::list state, py::object A,
+                                               int site,
+                                               const Strategy &strategy,
+                                               bool overwrite) {
   if (!is_array(A) || array_ndim(A) != 3) {
     throw std::invalid_argument(
-        "Invalid tensor passed to _update_in_canonical_form_right");
+        "Invalid tensor passed to _update_canonical_right");
   }
   py::object tensor = overwrite ? array_getcontiguous(A) : array_copy(A);
   auto a = array_dim(tensor, 0);
@@ -55,14 +56,12 @@ std::tuple<int, double> _update_in_canonical_form_left(py::list state,
 double _canonicalize(py::list state, int center, const Strategy &strategy) {
   double err = 0.0;
   for (int i = 0; i < center;) {
-    auto [site, errk] =
-        _update_in_canonical_form_right(state, state[i], i, strategy);
+    auto [site, errk] = _update_canonical_right(state, state[i], i, strategy);
     err += errk;
     i = site;
   }
   for (int i = state.size() - 1; i > center;) {
-    auto [site, errk] =
-        _update_in_canonical_form_left(state, state[i], i, strategy);
+    auto [site, errk] = _update_canonical_left(state, state[i], i, strategy);
     err += errk;
     i = site;
   }
diff --git a/tests/test_states/test_canonical.py b/tests/test_states/test_canonical.py
index a67db8e..74a579d 100644
--- a/tests/test_states/test_canonical.py
+++ b/tests/test_states/test_canonical.py
@@ -7,8 +7,8 @@
     random_uniform_mps,
 )
 from seemps.state.canonical_mps import (
-    _update_in_canonical_form_left,
-    _update_in_canonical_form_right,
+    _update_canonical_left,
+    _update_canonical_right,
     _canonicalize,
 )
 from ..fixture_mps_states import MPSStatesFixture
@@ -18,18 +18,18 @@
 class TestCanonicalForm(MPSStatesFixture):
     def test_local_update_canonical(self):
         #
-        # We verify that _update_in_canonical_form() leaves a tensor that
+        # We verify that _update_canonical() leaves a tensor that
         # is an approximate isometry.
         #
         def ok(Ψ, normalization=False):
             strategy = DEFAULT_STRATEGY.replace(normalize=normalization)
             for i in range(Ψ.size - 1):
                 ξ = Ψ.copy()
-                _update_in_canonical_form_right(ξ._data, ξ[i], i, strategy)
+                _update_canonical_right(ξ._data, ξ[i], i, strategy)
                 self.assertTrue(approximateIsometry(ξ[i], +1))
             for i in range(1, Ψ.size):
                 ξ = Ψ.copy()
-                _update_in_canonical_form_left(ξ._data, ξ[i], i, strategy)
+                _update_canonical_left(ξ._data, ξ[i], i, strategy)
                 self.assertTrue(approximateIsometry(ξ[i], -1))
 
         run_over_random_uniform_mps(ok)