diff --git a/src/tensors/adjoint.jl b/src/tensors/adjoint.jl
index ca484e77b..dfc1a4471 100644
--- a/src/tensors/adjoint.jl
+++ b/src/tensors/adjoint.jl
@@ -50,6 +50,8 @@ Base.@propagate_inbounds function subblock(t::AdjointTensorMap, (f₁, f₂)::Tu
     return permutedims(conj(data), (domainind(tp)..., codomainind(tp)...))
 end
 
+to_cpu(t::AdjointTensorMap) = adjoint(to_cpu(adjoint(t)))
+
 # Show
 #------
 function Base.showarg(io::IO, t::AdjointTensorMap, toplevel::Bool)
diff --git a/src/tensors/tensor.jl b/src/tensors/tensor.jl
index 300654632..526f4e489 100644
--- a/src/tensors/tensor.jl
+++ b/src/tensors/tensor.jl
@@ -532,7 +532,8 @@ end
 #---------------------------
 Base.convert(::Type{TensorMap}, t::TensorMap) = t
 function Base.convert(::Type{TensorMap}, t::AbstractTensorMap)
-    return copy!(TensorMap{scalartype(t)}(undef, space(t)), t)
+    A = storagetype(t)
+    return copy!(TensorMapWithStorage{scalartype(A), A}(undef, space(t)), t)
 end
 
 function Base.convert(::Type{TensorMapWithStorage{T, A}}, t::TensorMap) where {T, A}
diff --git a/test/cuda/tensors.jl b/test/cuda/tensors.jl
index e495eef08..a31015a4a 100644
--- a/test/cuda/tensors.jl
+++ b/test/cuda/tensors.jl
@@ -258,14 +258,15 @@ for V in spacelist
                 end
             end
         end
-        @timedtestset "Tensor conversion" begin # TODO adjoint conversion methods don't work yet
+        @timedtestset "Tensor conversion" begin
             W = V1 ⊗ V2
             t = @constinferred CUDA.randn(W ← W)
-            #@test typeof(convert(TensorMap, t')) == typeof(t) # TODO Adjoint not supported yet
+            @test typeof(convert(typeof(t), t')) == typeof(t)
+            @test typeof(TensorKit.to_cpu(t')) == typeof(TensorKit.to_cpu(t)')
             tc = complex(t)
             @test convert(typeof(tc), t) == tc
             @test typeof(convert(typeof(tc), t)) == typeof(tc)
-            # @test typeof(convert(typeof(tc), t')) == typeof(tc) # TODO Adjoint not supported yet
+            @test typeof(convert(typeof(tc), t')) == typeof(tc)
             @test Base.promote_typeof(t, tc) == typeof(tc)
             @test Base.promote_typeof(tc, t) == typeof(tc + t)
         end