diff --git a/src/spaces/homspace.jl b/src/spaces/homspace.jl
index 5f6d62a1..c1c4e6af 100644
--- a/src/spaces/homspace.jl
+++ b/src/spaces/homspace.jl
@@ -124,7 +124,13 @@ function permute(W::HomSpace{S}, (p₁, p₂)::Index2Tuple{N₁,N₂}) where {S,
     return cod ← dom
 end
 
-function Base.:*(W::HomSpace{S}, V::HomSpace{S}) where {S}
+"""
+    compose(W::HomSpace, V::HomSpace)
+
+Obtain the HomSpace that is obtained from composing the morphisms in `W` and `V`. For this
+to be possible, the domain of `W` must match the codomain of `V`.
+"""
+function compose(W::HomSpace{S}, V::HomSpace{S}) where {S}
     domain(W) == codomain(V) || throw(SpaceMismatch("$(domain(W)) ≠ $(codomain(V))"))
     return HomSpace(codomain(W), domain(V))
 end
diff --git a/src/tensors/linalg.jl b/src/tensors/linalg.jl
index 7c75a932..e00b1ee7 100644
--- a/src/tensors/linalg.jl
+++ b/src/tensors/linalg.jl
@@ -20,7 +20,7 @@ LinearAlgebra.normalize(t::AbstractTensorMap, p::Real=2) = scale(t, inv(norm(t,
 
 function Base.:*(t1::AbstractTensorMap, t2::AbstractTensorMap)
     return mul!(similar(t1, promote_type(scalartype(t1), scalartype(t2)),
-                        space(t1) * space(t2)), t1, t2)
+                        compose(space(t1), space(t2))), t1, t2)
 end
 Base.exp(t::AbstractTensorMap) = exp!(copy(t))
 function Base.:^(t::AbstractTensorMap, p::Integer)
@@ -242,7 +242,7 @@ end
 function LinearAlgebra.mul!(tC::AbstractTensorMap,
                             tA::AbstractTensorMap,
                             tB::AbstractTensorMap, α=true, β=false)
-    space(tA) * space(tB) == space(tC) ||
+    compose(space(tA), space(tB)) == space(tC) ||
         throw(SpaceMismatch("$(space(tC)) ≠ $(space(tA)) * $(space(tB))"))
 
     for c in blocksectors(tC)
diff --git a/src/tensors/tensoroperations.jl b/src/tensors/tensoroperations.jl
index ed084a4d..a2b6f171 100644
--- a/src/tensors/tensoroperations.jl
+++ b/src/tensors/tensoroperations.jl
@@ -128,7 +128,7 @@ function TO.tensorcontract_structure(pC::Index2Tuple{N₁,N₂},
                                      conjB) where {S,N₁,N₂}
     sA = TO.tensoradd_structure(pA, A, conjA)
     sB = TO.tensoradd_structure(pB, B, conjB)
-    return permute(sA * sB, pC)
+    return permute(compose(sA, sB), pC)
 end
 
 function TO.checkcontractible(tA::AbstractTensorMap{S}, iA::Int, conjA::Symbol,
diff --git a/test/spaces.jl b/test/spaces.jl
index 48b9e51d..42f4b922 100644
--- a/test/spaces.jl
+++ b/test/spaces.jl
@@ -412,6 +412,6 @@ println("------------------------------------")
         @test W == deepcopy(W)
         @test W == @constinferred permute(W, ((1, 2), (3, 4, 5)))
         @test permute(W, ((2, 4, 5), (3, 1))) == (V2 ⊗ V4' ⊗ V5' ← V3 ⊗ V1')
-        @test (V1 ⊗ V2 ← V1 ⊗ V2) == @constinferred W * W'
+        @test (V1 ⊗ V2 ← V1 ⊗ V2) == @constinferred compose(W, W')
     end
 end