diff --git a/src/Rings/MPolyMap/Types.jl b/src/Rings/MPolyMap/Types.jl
index 9f22448a96e..6c56fd1cf62 100644
--- a/src/Rings/MPolyMap/Types.jl
+++ b/src/Rings/MPolyMap/Types.jl
@@ -35,12 +35,15 @@ const _DomainTypes = Union{MPolyRing, MPolyQuoRing}
     # can safely be disabled.
     if check_for_mapping_of_vars && all(_is_gen, img_gens) && _allunique(img_gens)
       gens_codomain = gens(codomain)
-      result.variable_indices = [findfirst(==(x), gens_codomain) for x in img_gens]
+      result.variable_indices = [findfirst(_cmp_reps(x), gens_codomain) for x in img_gens]
     end
     return result
   end
 end
 
+# This can be overwritten in order to avoid making the above check a bottleneck.
+_cmp_reps(a) = ==(a)
+
 function MPolyAnyMap(d::D, c::C, cm::U, ig::Vector{V}) where {D, C, U, V}
   return MPolyAnyMap{D, C, U, V}(d, c, cm, ig)
 end
diff --git a/src/Rings/mpolyquo-localizations.jl b/src/Rings/mpolyquo-localizations.jl
index f9357af03df..e0e3ecf557d 100644
--- a/src/Rings/mpolyquo-localizations.jl
+++ b/src/Rings/mpolyquo-localizations.jl
@@ -2804,3 +2804,7 @@ _exponents(x::MPolyQuoRingElem) = AbstractAlgebra.exponent_vectors(lift(x))
 _exponents(x::MPolyLocRingElem) = AbstractAlgebra.exponent_vectors(numerator(x))
 _exponents(x::MPolyQuoLocRingElem) = AbstractAlgebra.exponent_vectors(lifted_numerator(x))
 
+# overwriting the comparison method to avoid computing saturations and groebner bases.
+_cmp_reps(a::MPolyLocRingElem) = y->(fraction(y) == fraction(a))
+_cmp_reps(a::MPolyQuoLocRingElem) = y->(fraction(y) == fraction(a))
+