diff --git a/Project.toml b/Project.toml
index a2afa92..0a9e304 100644
--- a/Project.toml
+++ b/Project.toml
@@ -2,7 +2,7 @@ name = "AlgebraicRewriting"
 uuid = "725a01d3-f174-5bbd-84e1-b9417bad95d9"
 license = "MIT"
 authors = ["Kris Brown <kris@topos.institute>"]
-version = "0.4.1"
+version = "0.4.2"
 
 [deps]
 ACSets = "227ef7b5-1206-438b-ac65-934d6da304b8"
diff --git a/src/incremental/Algorithms.jl b/src/incremental/Algorithms.jl
index a747889..664fc94 100644
--- a/src/incremental/Algorithms.jl
+++ b/src/incremental/Algorithms.jl
@@ -3,6 +3,7 @@ module Algorithms
 using Catlab 
 using ...CategoricalAlgebra.CSets: invert_iso, var_reference
 using ...Rewrite.Migration: pres_hash
+using ACSets.DenseACSets: attrtype_type
 
 """
 Break an ACSet into connected components, represented as a coproduct and an 
@@ -175,7 +176,7 @@ function nac_overlap(nac, update::ACSetTransformation)
 end 
 
 """
-Given f: L->X and m: X' ↣ X, find the unique map L -> X' making the triangle 
+Given f: X' ↣ X, m: L->X , find the unique map L -> X' making the triangle 
 commute, if it exists.
 
 TODO rewrite with @comptime
@@ -185,11 +186,12 @@ function pull_back(f::ACSetTransformation, m::ACSetTransformation
   L, X′ = dom.([m, f])
   comps, S = Dict(), acset_schema(L)
   for o in ob(S)
-    vec = []
+    vec = Int[]
     for i in parts(L, o)
       pre = preimage(f[o], m[o](i))
+      # Given that `m` is monic, preimage is empty or singleton
       length(pre) == 1 || return nothing
-      all(attrs(S; from=o)) do (atr, _, _)
+      all(attrs(S; from=o, just_names=true)) do atr
         L[i, atr] isa AttrVar || L[i,atr] == X′[only(pre), atr]
       end || return nothing
       push!(vec, only(pre))
@@ -199,6 +201,7 @@ function pull_back(f::ACSetTransformation, m::ACSetTransformation
   # Check that attribute variables in L are mapped consistently
   # i.e. L->X' doesn't send the variable to two different values.
   for o in attrtypes(S)
+    T = attrtype_type(L, o)
     comps[o] = map(AttrVar.(parts(L, o))) do i 
       vals = Set()
       for (f, c, _) in attrs(S; to=o)
@@ -207,9 +210,9 @@ function pull_back(f::ACSetTransformation, m::ACSetTransformation
         end
       end
       return only(vals)
-    end
+    end |> Vector{T}
   end
-  ACSetTransformation(dom(m), dom(f); comps...)
+  ACSetTransformation(L, X′; comps...)
 end
 
 """Get the pairs for each component of the image and its component"""
diff --git a/test/incremental/Algorithms.jl b/test/incremental/Algorithms.jl
index f71b037..5b16bb5 100644
--- a/test/incremental/Algorithms.jl
+++ b/test/incremental/Algorithms.jl
@@ -4,7 +4,7 @@ using Test
 using Catlab
 using AlgebraicRewriting.Incremental.Algorithms: 
   connected_acset_components, all_subobjects, subobject_cache, 
-  to_subobj, deattr
+  to_subobj, deattr, pull_back
 
 # test connected_acset_components
 #--------------------------------
@@ -37,5 +37,23 @@ G = SymmetricGraph(3) ⊕ cycle_graph(SymmetricGraph, 3)
 #--------
 @test deattr(SchWeightedGraph) == SchGraph
 
+# Pull_back
+#----------
+@present SchAttr(FreeSchema) begin 
+  X::Ob; D::AttrType; f::Attr(X,D); Y::Ob
+end
+@acset_type Attr(SchAttr)
+D1 = @acset Attr{Symbol} begin X=1; f=[AttrVar(1)]; D=1 end
+DXY = @acset Attr{Symbol} begin X=2; f=[:X,:Y] end
+DX = @acset Attr{Symbol} begin X=1; f=[:X] end
+toX,toY = homomorphisms(D1,DXY)
+subX = homomorphism(DX,DXY)
+toSubX = homomorphism(D1, DX)
+
+pb = pull_back(subX, toX)
+@test dom(pb) == dom(toX)
+@test codom(pb) == dom(subX)
+@test pb == toSubX
+@test isnothing(pull_back(subX, toY))
 
 end # module
diff --git a/test/runtests.jl b/test/runtests.jl
index f2ad5ad..88aab6b 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -13,7 +13,7 @@ using DataMigrations
 #######
 
 @testset "Full Demo" begin
-  # include("../docs/literate/full_demo.jl")
+  include("../docs/literate/full_demo.jl")
 end
 
 @testset "Lotka Volterra" begin
@@ -21,7 +21,7 @@ end
 end
 
 @testset "Game of Life" begin
-  include("../docs/literate/game_of_life.jl")
+  # include("../docs/literate/game_of_life.jl")
 end
 
 # Background 
diff --git a/test/schedules/Eval.jl b/test/schedules/Eval.jl
index 0530172..90dc64e 100644
--- a/test/schedules/Eval.jl
+++ b/test/schedules/Eval.jl
@@ -47,14 +47,14 @@ de = loop_rule(RuleApp(:del_edge, Rule(g2ar, id(g2))))
 coin = uniform(2, z)
 sched = coin ⋅ (tryrule(av) ⊗ id([z])) ⋅ merge_wires(z) ⋅ de
 
-view_sched(sched, name="Simple schedule", names=N)
+# view_sched(sched, name="Simple schedule", names=N)
 G = path_graph(Grph, 2)
 
 @test interpret!(sched, G) isa ACSetTransformation
 @test ne(G) == 0
 
 res = interpret(sched, path_graph(Grph, 2))
-view_traj(sched, res, view_graph; agent=true, names=N)
+# view_traj(sched, res, view_graph; agent=true, names=N)
 
 # Query workflow (add loop to each vertex)
 ##########################################
@@ -83,9 +83,9 @@ sched = mk_sched((o=Dot,), (i=:Z,), N, Dict(:rule=>al, :query=>q),
 
 typecheck(sched)
 
-view_sched(sched; names=N)
+# view_sched(sched; names=N)
 res = interpret(sched, Grph(3));
-view_traj(sched, res, view_graph; agent=true, names=N)
+# view_traj(sched, res, view_graph; agent=true, names=N)
 
 
 # Dependent query workflow 
@@ -111,18 +111,18 @@ quote
   return [trace1, trace2], out
 end);
 
-view_sched(sched; names=N)
+# view_sched(sched; names=N)
 
 G = @acset Grph begin V=5; E=4; src=[1,2,2,5];tgt=[2,3,4,2] end
 arr_start = homomorphism(ar, G; initial=(V=[1,2],))
 res = interpret(sched, arr_start);
-view_traj(sched, res, view_graph; agent=true, names=N)
+# view_traj(sched, res, view_graph; agent=true, names=N)
 @test interpret!(sched, arr_start) isa ACSetTransformation
 
 # For-loop: add 3 loops
 #######################
 sched = for_schedule(maybe_add_loop ⋅ merge_wires(g1), 3);
-view_sched(sched)
+# view_sched(sched)
 interpret!(sched, id(g1)) |> codom
 
 # TODO add an attributed example