diff --git a/Project.toml b/Project.toml
index 347c4c51..3f04c82b 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "ChainRulesTestUtils"
 uuid = "cdddcdb0-9152-4a09-a978-84456f9df70a"
-version = "0.5.7"
+version = "0.5.8"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
diff --git a/src/check_result.jl b/src/check_result.jl
index 683d7416..6be1af04 100644
--- a/src/check_result.jl
+++ b/src/check_result.jl
@@ -25,6 +25,10 @@ for (T1, T2) in ((AbstractThunk, Any), (AbstractThunk, AbstractThunk), (Any, Abs
     end
 end
 
+check_equal(::Zero, x; kwargs...) = check_equal(zero(x), x; kwargs...)
+check_equal(x, ::Zero; kwargs...) = check_equal(x, zero(x); kwargs...)
+check_equal(x::Zero, y::Zero; kwargs...) = @test true
+
 """
     _can_pass_early(actual, expected; kwargs...)
 Used to check if `actual` is basically equal to `expected`, so we don't need to check deeper;
@@ -77,16 +81,29 @@ function check_equal(
     @test ActualPrimal === ExpectedPrimal
 end
 
+
+# Some structual differential and a natural differential
+function check_equal(actual::Composite{P, T}, expected; kwargs...) where {T, P}
+    if _can_pass_early(actual, expected)
+        @test true
+    else
+        @assert (T <: NamedTuple)  # it should be a structual differential if we hit this
+
+        # We are only checking the properties that are in the Composite
+        # the natural differential is allowed to have other properties that we ignore
+        @testset "$P.$ii" for ii in propertynames(actual)
+            check_equal(getproperty(actual, ii), getproperty(expected, ii); kwargs...)
+        end
+    end
+end
+check_equal(x, y::Composite; kwargs...) = check_equal(y, x; kwargs...)
+
 # This catches comparisons of Composites and Tuples/NamedTuple
 # and gives a error messaage complaining about that
 const LegacyZygoteCompTypes = Union{Tuple,NamedTuple}
 check_equal(::C, expected::T) where {C<:Composite,T<:LegacyZygoteCompTypes} = @test C === T
 check_equal(::T, expected::C) where {C<:Composite,T<:LegacyZygoteCompTypes} = @test T === C
 
-check_equal(::Zero, x; kwargs...) = check_equal(zero(x), x; kwargs...)
-check_equal(x, ::Zero; kwargs...) = check_equal(x, zero(x); kwargs...)
-check_equal(x::Zero, y::Zero; kwargs...) = @test true
-
 # Generic fallback, probably a tuple or something
 function check_equal(actual::A, expected::E; kwargs...) where {A, E}
     if _can_pass_early(actual, expected)
@@ -101,6 +118,7 @@ function check_equal(actual::A, expected::E; kwargs...) where {A, E}
     end
 end
 
+
 """
 _check_add!!_behavour(acc, val)
 
diff --git a/test/check_result.jl b/test/check_result.jl
index 1e98155e..5960447c 100644
--- a/test/check_result.jl
+++ b/test/check_result.jl
@@ -51,10 +51,14 @@ end
                 Composite{Tuple{Float64, Float64}}(1.0, 2.0)
             )
 
-            D = Diagonal(randn(5))
-            check_equal(
-                Composite{typeof(D)}(diag=D.diag),
-                Composite{typeof(D)}(diag=D.diag)
+            diag_eg = Diagonal(randn(5))
+            check_equal( # Structual == Structural
+                Composite{typeof(diag_eg)}(diag=diag_eg.diag),
+                Composite{typeof(diag_eg)}(diag=diag_eg.diag)
+            )
+            check_equal( # Structural == Natural
+                Composite{typeof(diag_eg)}(diag=diag_eg.diag),
+                diag_eg
             )
 
             T = (a=1.0, b=2.0)