diff --git a/src/spaces/cartesianspace.jl b/src/spaces/cartesianspace.jl
index 87822d59..dbe1a534 100644
--- a/src/spaces/cartesianspace.jl
+++ b/src/spaces/cartesianspace.jl
@@ -27,6 +27,7 @@ function CartesianSpace(dims::AbstractDict; kwargs...)
         throw(SectorMismatch(msg))
     end
 end
+CartesianSpace(g::Base.Generator; kwargs...) = CartesianSpace(g...; kwargs...)
 
 field(::Type{CartesianSpace}) = ℝ
 InnerProductStyle(::Type{CartesianSpace}) = EuclideanProduct()
diff --git a/src/spaces/complexspace.jl b/src/spaces/complexspace.jl
index a9929f9a..d266744b 100644
--- a/src/spaces/complexspace.jl
+++ b/src/spaces/complexspace.jl
@@ -28,6 +28,7 @@ function ComplexSpace(dims::AbstractDict; kwargs...)
         throw(SectorMismatch(msg))
     end
 end
+ComplexSpace(g::Base.Generator; kwargs...) = ComplexSpace(g...; kwargs...)
 
 field(::Type{ComplexSpace}) = ℂ
 InnerProductStyle(::Type{ComplexSpace}) = EuclideanProduct()
diff --git a/src/spaces/homspace.jl b/src/spaces/homspace.jl
index abcfb5ef..c1c4e6af 100644
--- a/src/spaces/homspace.jl
+++ b/src/spaces/homspace.jl
@@ -115,3 +115,22 @@ function dim(W::HomSpace)
     end
     return d
 end
+
+# Operations on HomSpaces
+# -----------------------
+function permute(W::HomSpace{S}, (p₁, p₂)::Index2Tuple{N₁,N₂}) where {S,N₁,N₂}
+    cod = ProductSpace{S,N₁}(map(n -> W[n], p₁))
+    dom = ProductSpace{S,N₂}(map(n -> dual(W[n]), p₂))
+    return cod ← dom
+end
+
+"""
+    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/spaces/vectorspaces.jl b/src/spaces/vectorspaces.jl
index b2b7a454..4155e743 100644
--- a/src/spaces/vectorspaces.jl
+++ b/src/spaces/vectorspaces.jl
@@ -160,6 +160,7 @@ individual spaces `V₁`, `V₂`, ..., or the spaces contained in `P`.
 """
 function fuse end
 fuse(V::ElementarySpace) = isdual(V) ? flip(V) : V
+fuse(V::ElementarySpace, W::ElementarySpace) = fuse(promote(V, W)...)
 function fuse(V₁::VectorSpace, V₂::VectorSpace, V₃::VectorSpace...)
     return fuse(fuse(fuse(V₁), fuse(V₂)), V₃...)
 end
diff --git a/src/tensors/indexmanipulations.jl b/src/tensors/indexmanipulations.jl
index dccb38a2..0c202245 100644
--- a/src/tensors/indexmanipulations.jl
+++ b/src/tensors/indexmanipulations.jl
@@ -30,19 +30,19 @@ To permute into an existing destination, see [permute!](@ref) and [`add_permute!
 """
 function permute(t::AbstractTensorMap{S}, (p₁, p₂)::Index2Tuple{N₁,N₂};
                  copy::Bool=false) where {S,N₁,N₂}
-    cod = ProductSpace{S,N₁}(map(n -> space(t, n), p₁))
-    dom = ProductSpace{S,N₂}(map(n -> dual(space(t, n)), p₂))
+    space′ = permute(space(t), (p₁, p₂))
     # share data if possible
     if !copy
         if p₁ === codomainind(t) && p₂ === domainind(t)
             return t
         elseif has_shared_permute(t, (p₁, p₂))
-            return TensorMap(reshape(t.data, dim(cod), dim(dom)), cod, dom)
+            return TensorMap(reshape(t.data, dim(codomain(space′)), dim(domain(space′))),
+                             codomain(space′), domain(space′))
         end
     end
     # general case
     @inbounds begin
-        return permute!(similar(t, cod ← dom), t, (p₁, p₂))
+        return permute!(similar(t, space′), t, (p₁, p₂))
     end
 end
 function permute(t::AdjointTensorMap{S}, (p₁, p₂)::Index2Tuple;
@@ -118,10 +118,9 @@ function braid(t::AbstractTensorMap{S}, (p₁, p₂)::Index2Tuple, levels::Index
         return t
     end
     # general case
-    cod = ProductSpace{S}(map(n -> space(t, n), p₁))
-    dom = ProductSpace{S}(map(n -> dual(space(t, n)), p₂))
+    space′ = permute(space(t), (p₁, p₂))
     @inbounds begin
-        return braid!(similar(t, cod ← dom), t, (p₁, p₂), levels)
+        return braid!(similar(t, space′), t, (p₁, p₂), levels)
     end
 end
 # TODO: braid for `AdjointTensorMap`; think about how to map the `levels` argument.
@@ -171,10 +170,9 @@ function LinearAlgebra.transpose(t::AbstractTensorMap{S},
         return t
     end
     # general case
-    cod = ProductSpace{S}(map(n -> space(t, n), p₁))
-    dom = ProductSpace{S}(map(n -> dual(space(t, n)), p₂))
+    space′ = permute(space(t), (p₁, p₂))
     @inbounds begin
-        return transpose!(similar(t, cod ← dom), t, (p₁, p₂))
+        return transpose!(similar(t, space′), t, (p₁, p₂))
     end
 end
 
@@ -313,10 +311,7 @@ function add_transform!(tdst::AbstractTensorMap{S,N₁,N₂},
                         β::Number,
                         backend::Backend...) where {S,N₁,N₂}
     @boundscheck begin
-        all(i -> space(tsrc, p₁[i]) == space(tdst, i), 1:N₁) ||
-            throw(SpaceMismatch("source = $(codomain(tsrc))←$(domain(tsrc)),
-            dest = $(codomain(tdst))←$(domain(tdst)), p₁ = $(p₁), p₂ = $(p₂)"))
-        all(i -> space(tsrc, p₂[i]) == space(tdst, N₁ + i), 1:N₂) ||
+        permute(space(tsrc), (p₁, p₂)) == space(tdst) ||
             throw(SpaceMismatch("source = $(codomain(tsrc))←$(domain(tsrc)),
             dest = $(codomain(tdst))←$(domain(tdst)), p₁ = $(p₁), p₂ = $(p₂)"))
     end
@@ -357,7 +352,7 @@ end
 
 function _add_abelian_block!(tdst, tsrc, p, fusiontreetransform, f₁, f₂, α, β, backend...)
     (f₁′, f₂′), coeff = first(fusiontreetransform(f₁, f₂))
-    TO.tensoradd!(tdst[f₁′, f₂′], p, tsrc[f₁, f₂], :N, α * coeff, β, backend...)
+    @inbounds TO.tensoradd!(tdst[f₁′, f₂′], p, tsrc[f₁, f₂], :N, α * coeff, β, backend...)
     return nothing
 end
 
@@ -375,8 +370,8 @@ function _add_general_kernel!(tdst, tsrc, p, fusiontreetransform, α, β, backen
     else
         for (f₁, f₂) in fusiontrees(tsrc)
             for ((f₁′, f₂′), coeff) in fusiontreetransform(f₁, f₂)
-                TO.tensoradd!(tdst[f₁′, f₂′], p, tsrc[f₁, f₂], :N, α * coeff, true,
-                              backend...)
+                @inbounds TO.tensoradd!(tdst[f₁′, f₂′], p, tsrc[f₁, f₂], :N, α * coeff,
+                                        true, backend...)
             end
         end
     end
@@ -388,7 +383,8 @@ function _add_nonabelian_sector!(tdst, tsrc, p, fusiontreetransform, s₁, s₂,
     for (f₁, f₂) in fusiontrees(tsrc)
         (f₁.uncoupled == s₁ && f₂.uncoupled == s₂) || continue
         for ((f₁′, f₂′), coeff) in fusiontreetransform(f₁, f₂)
-            TO.tensoradd!(tdst[f₁′, f₂′], p, tsrc[f₁, f₂], :N, α * coeff, true, backend...)
+            @inbounds TO.tensoradd!(tdst[f₁′, f₂′], p, tsrc[f₁, f₂], :N, α * coeff, true,
+                                    backend...)
         end
     end
     return nothing
diff --git a/src/tensors/linalg.jl b/src/tensors/linalg.jl
index 4678b780..46bbb285 100644
--- a/src/tensors/linalg.jl
+++ b/src/tensors/linalg.jl
@@ -18,10 +18,18 @@ Base.:\(α::Number, t::AbstractTensorMap) = *(t, one(scalartype(t)) / α)
 LinearAlgebra.normalize!(t::AbstractTensorMap, p::Real=2) = scale!(t, inv(norm(t, p)))
 LinearAlgebra.normalize(t::AbstractTensorMap, p::Real=2) = scale(t, inv(norm(t, p)))
 
-function Base.:*(t1::AbstractTensorMap, t2::AbstractTensorMap)
+"""
+    compose(t1::AbstractTensorMap, t2::AbstractTensorMap) -> AbstractTensorMap
+
+Return the `AbstractTensorMap` that implements the composition of the two tensor maps `t1`
+and `t2`.
+"""
+function compose(t1::AbstractTensorMap, t2::AbstractTensorMap)
     return mul!(similar(t1, promote_type(scalartype(t1), scalartype(t2)),
-                        codomain(t1) ← domain(t2)), t1, t2)
+                        compose(space(t1), space(t2))), t1, t2)
 end
+Base.:*(t1::AbstractTensorMap, t2::AbstractTensorMap) = compose(t1, t2)
+
 Base.exp(t::AbstractTensorMap) = exp!(copy(t))
 function Base.:^(t::AbstractTensorMap, p::Integer)
     return p < 0 ? Base.power_by_squaring(inv(t), -p) : Base.power_by_squaring(t, p)
@@ -242,10 +250,9 @@ end
 function LinearAlgebra.mul!(tC::AbstractTensorMap,
                             tA::AbstractTensorMap,
                             tB::AbstractTensorMap, α=true, β=false)
-    if !(codomain(tC) == codomain(tA) && domain(tC) == domain(tB) &&
-         domain(tA) == codomain(tB))
+    compose(space(tA), space(tB)) == space(tC) ||
         throw(SpaceMismatch("$(space(tC)) ≠ $(space(tA)) * $(space(tB))"))
-    end
+
     for c in blocksectors(tC)
         if hasblock(tA, c) # then also tB should have such a block
             A = block(tA, c)
diff --git a/src/tensors/tensoroperations.jl b/src/tensors/tensoroperations.jl
index 9f8ed5a9..a2b6f171 100644
--- a/src/tensors/tensoroperations.jl
+++ b/src/tensors/tensoroperations.jl
@@ -59,9 +59,7 @@ end
 function TO.tensoradd_structure(pC::Index2Tuple{N₁,N₂},
                                 A::AbstractTensorMap{S}, conjA::Symbol) where {S,N₁,N₂}
     if conjA == :N
-        cod = ProductSpace{S,N₁}(space.(Ref(A), pC[1]))
-        dom = ProductSpace{S,N₂}(dual.(space.(Ref(A), pC[2])))
-        return dom → cod
+        return permute(space(A), pC)
     else
         return TO.tensoradd_structure(adjointtensorindices(A, pC), adjoint(A), :N)
     end
@@ -128,12 +126,9 @@ function TO.tensorcontract_structure(pC::Index2Tuple{N₁,N₂},
                                      A::AbstractTensorMap{S}, pA::Index2Tuple, conjA,
                                      B::AbstractTensorMap{S}, pB::Index2Tuple,
                                      conjB) where {S,N₁,N₂}
-    spaces1 = TO.flag2op(conjA).(space.(Ref(A), pA[1]))
-    spaces2 = TO.flag2op(conjB).(space.(Ref(B), pB[2]))
-    spaces = (spaces1..., spaces2...)
-    cod = ProductSpace{S,N₁}(getindex.(Ref(spaces), pC[1]))
-    dom = ProductSpace{S,N₂}(dual.(getindex.(Ref(spaces), pC[2])))
-    return dom → cod
+    sA = TO.tensoradd_structure(pA, A, conjA)
+    sB = TO.tensoradd_structure(pB, B, conjB)
+    return permute(compose(sA, sB), pC)
 end
 
 function TO.checkcontractible(tA::AbstractTensorMap{S}, iA::Int, conjA::Symbol,
@@ -165,10 +160,7 @@ function trace_permute!(tdst::AbstractTensorMap{S,N₁,N₂},
         throw(SectorMismatch("only tensors with symmetric braiding rules can be contracted; try `@planar` instead"))
     end
     @boundscheck begin
-        all(i -> space(tsrc, p₁[i]) == space(tdst, i), 1:N₁) ||
-            throw(SpaceMismatch("trace: tsrc = $(codomain(tsrc))←$(domain(tsrc)),
-                    tdst = $(codomain(tdst))←$(domain(tdst)), p₁ = $(p₁), p₂ = $(p₂)"))
-        all(i -> space(tsrc, p₂[i]) == space(tdst, N₁ + i), 1:N₂) ||
+        space(tdst) == permute(space(tsrc), (p₁, p₂)) ||
             throw(SpaceMismatch("trace: tsrc = $(codomain(tsrc))←$(domain(tsrc)),
                     tdst = $(codomain(tdst))←$(domain(tdst)), p₁ = $(p₁), p₂ = $(p₂)"))
         all(i -> space(tsrc, q₁[i]) == dual(space(tsrc, q₂[i])), 1:N₃) ||
diff --git a/test/spaces.jl b/test/spaces.jl
index e2171d0a..5b591bca 100644
--- a/test/spaces.jl
+++ b/test/spaces.jl
@@ -410,5 +410,8 @@ println("------------------------------------")
         @test W[5] == V5'
         @test @constinferred(hash(W)) == hash(deepcopy(W)) != hash(W')
         @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 TensorKit.compose(W, W')
     end
 end