diff --git a/src/Utilities/matrix_of_constraints.jl b/src/Utilities/matrix_of_constraints.jl
index 921dd820ac..797aa8ef55 100644
--- a/src/Utilities/matrix_of_constraints.jl
+++ b/src/Utilities/matrix_of_constraints.jl
@@ -92,13 +92,24 @@ mutable struct MatrixOfConstraints{T,AT,BT,ST} <: MOI.ModelLike
     caches::Vector{Any}
     are_indices_mapped::Vector{BitSet}
     final_touch::Bool
-    function MatrixOfConstraints{T,AT,BT,ST}() where {T,AT,BT,ST}
-        model = new{T,AT,BT,ST}(AT(), BT(), ST(), Any[], BitSet[], false)
-        MOI.empty!(model)
+    function MatrixOfConstraints{T}(coefficients, constants, sets) where {T}
+        model = new{T,typeof(coefficients),typeof(constants),typeof(sets)}(
+            coefficients,
+            constants,
+            sets,
+            Any[],
+            BitSet[],
+            false,
+        )
+        _reset_caches!(model)
         return model
     end
 end
 
+function MatrixOfConstraints{T,AT,BT,ST}() where {T,AT,BT,ST}
+    return MatrixOfConstraints{T}(AT(), BT(), ST())
+end
+
 ###
 ### Interface for the .coefficients field
 ###
@@ -292,13 +303,17 @@ function rows end
 
 MOI.is_empty(v::MatrixOfConstraints) = MOI.is_empty(v.sets)
 
-function MOI.empty!(v::MatrixOfConstraints{T}) where {T}
+function _reset_caches!(v::MatrixOfConstraints{T}) where {T}
+    v.caches =
+        [Tuple{_affine_function_type(T, S),S}[] for S in set_types(v.sets)]
+    return v.are_indices_mapped = [BitSet() for _ in eachindex(v.caches)]
+end
+
+function MOI.empty!(v::MatrixOfConstraints)
     MOI.empty!(v.coefficients)
     empty!(v.constants)
     MOI.empty!(v.sets)
-    v.caches =
-        [Tuple{_affine_function_type(T, S),S}[] for S in set_types(v.sets)]
-    v.are_indices_mapped = [BitSet() for _ in eachindex(v.caches)]
+    _reset_caches!(v)
     v.final_touch = false
     return
 end
diff --git a/src/Utilities/model.jl b/src/Utilities/model.jl
index 184488c3b8..06ef912a3d 100644
--- a/src/Utilities/model.jl
+++ b/src/Utilities/model.jl
@@ -32,12 +32,12 @@ mutable struct GenericModel{T,O,V,C} <: AbstractModelLike{T}
     # A useful dictionary for extensions to store things. These are
     # _not_ copied between models.
     ext::Dict{Symbol,Any}
-    function GenericModel{T,O,V,C}() where {T,O,V,C}
-        return new{T,O,V,C}(
+    function GenericModel{T}(objective, variables, constraints) where {T}
+        return new{T,typeof(objective),typeof(variables),typeof(constraints)}(
             "",
-            O(),
-            V(),
-            C(),
+            objective,
+            variables,
+            constraints,
             Dict{MOI.VariableIndex,String}(),
             nothing,
             Dict{MOI.ConstraintIndex,String}(),
@@ -47,6 +47,10 @@ mutable struct GenericModel{T,O,V,C} <: AbstractModelLike{T}
     end
 end
 
+function GenericModel{T,O,V,C}() where {T,O,V,C}
+    return GenericModel{T}(O(), V(), C())
+end
+
 abstract type AbstractOptimizer{T} <: MOI.AbstractOptimizer end
 
 """
diff --git a/src/Utilities/sparse_matrix.jl b/src/Utilities/sparse_matrix.jl
index 98b4b4195c..fd5b37f41a 100644
--- a/src/Utilities/sparse_matrix.jl
+++ b/src/Utilities/sparse_matrix.jl
@@ -183,19 +183,21 @@ function Base.convert(
     )
 end
 
-function _first_in_column(
-    A::MutableSparseMatrixCSC{Tv,Ti},
-    row::Integer,
-    col::Integer,
-) where {Tv,Ti}
+_indexing(A::MutableSparseMatrixCSC) = A.indexing
+_indexing(::SparseArrays.SparseMatrixCSC) = OneBasedIndexing()
+
+const _SparseMatrixCSC{Tv,Ti} =
+    Union{MutableSparseMatrixCSC{Tv,Ti},SparseArrays.SparseMatrixCSC{Tv,Ti}}
+
+function _first_in_column(A::_SparseMatrixCSC, row::Integer, col::Integer)
     range = SparseArrays.nzrange(A, col)
-    row = _shift(row, OneBasedIndexing(), A.indexing)
+    row = _shift(row, OneBasedIndexing(), _indexing(A))
     idx = searchsortedfirst(view(A.rowval, range), row)
     return get(range, idx, last(range) + 1)
 end
 
 function extract_function(
-    A::MutableSparseMatrixCSC{T},
+    A::_SparseMatrixCSC{T},
     row::Integer,
     constant::T,
 ) where {T}
@@ -205,7 +207,7 @@ function extract_function(
         if idx > last(SparseArrays.nzrange(A, col))
             continue
         end
-        r = _shift(A.rowval[idx], A.indexing, OneBasedIndexing())
+        r = _shift(A.rowval[idx], _indexing(A), OneBasedIndexing())
         if r == row
             push!(
                 func.terms,
@@ -217,7 +219,7 @@ function extract_function(
 end
 
 function extract_function(
-    A::MutableSparseMatrixCSC{T},
+    A::_SparseMatrixCSC{T},
     rows::UnitRange,
     constants::Vector{T},
 ) where {T}
@@ -231,7 +233,7 @@ function extract_function(
             if idx[col] > last(SparseArrays.nzrange(A, col))
                 continue
             end
-            row = _shift(A.rowval[idx[col]], A.indexing, OneBasedIndexing())
+            row = _shift(A.rowval[idx[col]], _indexing(A), OneBasedIndexing())
             if row != rows[output_index]
                 continue
             end
diff --git a/test/Utilities/matrix_of_constraints.jl b/test/Utilities/matrix_of_constraints.jl
index 96069f4c63..fa2f405d17 100644
--- a/test/Utilities/matrix_of_constraints.jl
+++ b/test/Utilities/matrix_of_constraints.jl
@@ -714,6 +714,78 @@ function test_unsupported_constraint()
     return
 end
 
+MOI.Utilities.@product_of_sets(_EqualTos, MOI.EqualTo{T},)
+
+function _equality_constraints(
+    A::AbstractMatrix{T},
+    b::AbstractVector{T},
+) where {T}
+    sets = _EqualTos{T}()
+    for _ in eachindex(b)
+        MOI.Utilities.add_set(
+            sets,
+            MOI.Utilities.set_index(sets, MOI.EqualTo{T}),
+        )
+    end
+    MOI.Utilities.final_touch(sets)
+    constants = MOI.Utilities.Hyperrectangle(b, b)
+    model = MOI.Utilities.MatrixOfConstraints{T}(A, constants, sets)
+    model.final_touch = true
+    return model
+end
+
+# Inspired from MatrixOfConstraints
+function test_lp_standard_form()
+    s = """
+    variables: x1, x2
+    cx1: x1 >= 0.0
+    cx2: x2 >= 0.0
+    c1: 1x1       == 5.0
+    c2: 3x1 + 4x2 == 6.0
+    minobjective: 7x1 + 8x2
+    """
+    expected = MOI.Utilities.Model{Float64}()
+    MOI.Utilities.loadfromstring!(expected, s)
+
+    var_names = ["x1", "x2"]
+    con_names = ["c1", "c2"]
+
+    A = SparseArrays.sparse([
+        1.0 0.0
+        3.0 4.0
+    ])
+    b = [5.0, 6.0]
+    form = MOI.Utilities.GenericModel{Float16}(
+        expected.objective,
+        expected.variables,
+        _equality_constraints(A, b),
+    )
+
+    model = MOI.Utilities.Model{Float64}()
+    MOI.copy_to(MOI.Bridges.Constraint.Scalarize{Float64}(model), form)
+    MOI.set(
+        model,
+        MOI.VariableName(),
+        MOI.VariableIndex.(eachindex(var_names)),
+        var_names,
+    )
+    MOI.set(
+        model,
+        MOI.ConstraintName(),
+        MOI.ConstraintIndex{
+            MOI.ScalarAffineFunction{Float64},
+            MOI.EqualTo{Float64},
+        }.(
+            eachindex(con_names),
+        ),
+        con_names,
+    )
+
+    MOI.Test.util_test_models_equal(model, expected, var_names, con_names)
+
+    return
+end
+
 end
 
 TestMatrixOfConstraints.runtests()