--- a/src/LazyTensors/lazy_tensor_operations.jl	Mon Nov 02 21:33:35 2020 +0100
+++ b/src/LazyTensors/lazy_tensor_operations.jl	Thu Nov 05 11:32:49 2020 +0100
@@ -86,12 +86,9 @@
     t2::TM2
 
     @inline function TensorMappingComposition(t1::TensorMapping{T,R,K}, t2::TensorMapping{T,K,D}) where {T,R,K,D}
-        @boundscheck if domain_size(t1) != range_size(t2)
-            throw(DimensionMismatch("the first argument has domain size $(domain_size(t1)) while the second has range size $(range_size(t2)) "))
-        end
+        @boundscheck check_domain_size(t1, range_size(t2))
         return new{T,R,K,D, typeof(t1), typeof(t2)}(t1,t2)
     end
-    # Add check for matching sizes as a boundscheck
 end
 export TensorMappingComposition
 
@@ -170,6 +167,27 @@
 apply(tmi::IdentityMapping{T,D}, v::AbstractArray{T,D}, I::Vararg{Any,D}) where {T,D} = v[I...]
 apply_transpose(tmi::IdentityMapping{T,D}, v::AbstractArray{T,D}, I::Vararg{Any,D}) where {T,D} = v[I...]
 
+"""
+Base.:∘(tm, tmi)
+Base.:∘(tmi, tm)
+
+Composes a `Tensormapping` `tm` with an `IdentityMapping` `tmi`, by returning `tm`
+"""
+@inline function Base.:∘(tm::TensorMapping{T,R,D}, tmi::IdentityMapping{T,D}) where {T,R,D}
+    @boundscheck check_domain_size(tm, range_size(tmi))
+    return tm
+end
+
+@inline function Base.:∘(tmi::IdentityMapping{T,R}, tm::TensorMapping{T,R,D}) where {T,R,D}
+    @boundscheck check_domain_size(tmi, range_size(tm))
+    return tm
+end
+# Specialization for the case where tm is an IdentityMapping. Required to resolve ambiguity.
+@inline function Base.:∘(tm::IdentityMapping{T,D}, tmi::IdentityMapping{T,D}) where {T,D}
+    @boundscheck check_domain_size(tm, range_size(tmi))
+    return tmi
+end
+
 
 """
     InflatedTensorMapping{T,R,D} <: TensorMapping{T,R,D}
@@ -279,3 +297,20 @@
 flatten_tuple(t::NTuple{N, Number} where N) = t
 flatten_tuple(t::Tuple) = ((flatten_tuple.(t)...)...,) # simplify?
 flatten_tuple(ts::Vararg) = flatten_tuple(ts)
+
+function check_domain_size(tm::TensorMapping, sz)
+    if domain_size(tm) != sz
+        throw(SizeMismatch(tm,sz))
+    end
+end
+
+struct SizeMismatch <: Exception
+    tm::TensorMapping
+    sz
+end
+export SizeMismatch
+
+function Base.showerror(io::IO, err::SizeMismatch)
+    print(io, "SizeMismatch: ")
+    print(io, "domain size $(domain_size(err.tm)) of TensorMapping not matching size $(err.sz)")
+end

--- a/test/testLazyTensors.jl	Mon Nov 02 21:33:35 2020 +0100
+++ b/test/testLazyTensors.jl	Thu Nov 05 11:32:49 2020 +0100
@@ -177,6 +177,7 @@
     @test_throws BoundsError (v1 +̃  v2)[4]
     v2 = [1., 2, 3, 4]
     # Test that size of arrays is asserted when not specified inbounds
+    # TODO: Replace these errors with SizeMismatch
     @test_throws DimensionMismatch v1 +̃ v2
 
     # Test operations on LazyArray
@@ -193,6 +194,7 @@
     @test_throws BoundsError (v1 + v2)[4]
     v2 = [1., 2, 3, 4]
     # Test that size of arrays is asserted when not specified inbounds
+    # TODO: Replace these errors with SizeMismatch
     @test_throws DimensionMismatch v1 + v2
 end
 
@@ -226,7 +228,7 @@
     @test Ã∘B̃ isa TensorMappingComposition
     @test range_size(Ã∘B̃) == (2,)
     @test domain_size(Ã∘B̃) == (4,)
-    @test_throws DimensionMismatch B̃∘Ã
+    @test_throws SizeMismatch B̃∘Ã
 
     # @test @inbounds B̃∘Ã # Should not error even though dimensions don't match. (Since ]test runs with forced boundschecking this is currently not testable 2020-10-16)
 
@@ -312,6 +314,17 @@
 
     @inferred range_dim(I)
     @inferred domain_dim(I)
+
+    Ã = rand(4,2)
+    A = LazyLinearMap(Ã,(1,),(2,))
+    I1 = IdentityMapping{Float64}(2)
+    I2 = IdentityMapping{Float64}(4)
+    @test A∘I1 == A
+    @test I2∘A == A
+    @test I1∘I1 == I1
+    @test_throws SizeMismatch I1∘A
+    @test_throws SizeMismatch A∘I2
+    @test_throws SizeMismatch I1∘I2
 end
 
 @testset "InflatedTensorMapping" begin