diff --git a/docs/src/submodules/Bridges/reference.md b/docs/src/submodules/Bridges/reference.md
index 2cf3df3a9c..44d9a1855e 100644
--- a/docs/src/submodules/Bridges/reference.md
+++ b/docs/src/submodules/Bridges/reference.md
@@ -20,7 +20,7 @@ get(::Bridges.AbstractBridge, ::NumberOfConstraints)
 get(::Bridges.AbstractBridge, ::ListOfConstraintIndices)
 Bridges.needs_final_touch
 Bridges.final_touch
-Bridges.cost
+Bridges.bridging_cost
 ```
 
 ## Constraint bridge API
@@ -74,7 +74,6 @@ Bridges.unbridged_variable_function
 Bridges.bridged_function
 Bridges.supports_constraint_bridges
 Bridges.recursive_model
-Bridges.bridging_cost
 ```
 
 ## LazyBridgeOptimizer API
diff --git a/src/Bridges/Constraint/bridges/functionize.jl b/src/Bridges/Constraint/bridges/functionize.jl
index c517b8df13..9c4fcc1c02 100644
--- a/src/Bridges/Constraint/bridges/functionize.jl
+++ b/src/Bridges/Constraint/bridges/functionize.jl
@@ -202,7 +202,7 @@ end
     conversion_cost(
         F::Type{<:MOI.AbstractFunction},
         G::Type{<:MOI.AbstractFunction},
-    )::Int
+    )::Union{Int,Nothing}
 
 Return an `Int` returning the cost of converting any function of type `G`
 to a function of type `F` with `convert`.
@@ -273,7 +273,7 @@ function concrete_bridge_type(
     return FunctionConversionBridge{T,F,G,S}
 end
 
-function MOI.Bridges.cost(
+function MOI.Bridges.bridging_cost(
     ::Type{<:FunctionConversionBridge{T,F,G}},
 ) where {T,F,G}
     return conversion_cost(F, G)
diff --git a/src/Bridges/bridge.jl b/src/Bridges/bridge.jl
index 5c44800e6d..54d2cfb8a5 100644
--- a/src/Bridges/bridge.jl
+++ b/src/Bridges/bridge.jl
@@ -285,15 +285,13 @@ If you implement this method, you must also implement
 function final_touch end
 
 """
-    cost(BT::Type{<:AbstractBridge})::Int
+    bridging_cost(BT::Type{<:AbstractBridge})::Int
 
-Return the *cost* a bridge of type `BT`.
+Return the cost of adding a bridge of type `BT`.
 
-This cost is used to compute the [`bridging_cost`](@ref).
-
-There is a default implementation for any [`AbstractBridge`](@ref) returning
-`1` so this should only be implemented for bridges returning a distance
-different from `1`.
+The default implementation for any [`AbstractBridge`](@ref) returns `1`, so this
+method should only be implemented for bridges returning a cost different from
+`1`.
 
 ## Example
 
@@ -302,22 +300,23 @@ constraints from a given function type to a function type to a wider one,
 we want it to have lower priority.
 
 For example, we want to prioritize bridging a
-[`MOI.ScalarAffineFunction`](@ref)-in-[`MOI.LessThan`](@ref) constraint into
-a [`MOI.VectorAffineFunction`](@ref)-in-[`MOI.Nonnegatives`](@ref) constraint
-over bridging it to a [`MOI.ScalarQuadraticFunction`](@ref)-in-[`MOI.LessThan`](@ref) constraint.
+[`MOI.ScalarAffineFunction`](@ref)-in-[`MOI.LessThan`](@ref) constraint into a
+[`MOI.VectorAffineFunction`](@ref)-in-[`MOI.Nonnegatives`](@ref) constraint
+over bridging it to a [`MOI.ScalarQuadraticFunction`](@ref)-in-[`MOI.LessThan`](@ref)
+constraint.
 
 For this reason, the [`Bridges.Constraint.FunctionConversionBridge`](@ref) is
 given a cost of `10`:
 
 ```jldoctest; setup=(import MathOptInterface as MOI)
-julia> T = Float64;
-
-julia> F = MOI.ScalarQuadraticFunction{T};
+julia> F = MOI.ScalarQuadraticFunction{Float64};
 
-julia> G = MOI.ScalarAffineFunction{T};
+julia> G = MOI.ScalarAffineFunction{Float64};
 
-julia> MOI.Bridges.cost(MOI.Bridges.Constraint.FunctionConversionBridge{T,F,G})
+julia> MOI.Bridges.bridging_cost(
+           MOI.Bridges.Constraint.FunctionConversionBridge{Float64,F,G},
+       )
 10
 ```
 """
-cost(::Type{<:AbstractBridge}) = 1
+bridging_cost(::Type{<:AbstractBridge}) = 1
diff --git a/src/Bridges/lazy_bridge_optimizer.jl b/src/Bridges/lazy_bridge_optimizer.jl
index 5b809f430e..daac29de41 100644
--- a/src/Bridges/lazy_bridge_optimizer.jl
+++ b/src/Bridges/lazy_bridge_optimizer.jl
@@ -156,7 +156,7 @@ function _edge(b::LazyBridgeOptimizer, index::Int, BT::Type{<:AbstractBridge})
         index,
         _variable_nodes(b, BT),
         _constraint_nodes(b, BT),
-        cost(BT),
+        bridging_cost(BT),
     )
 end
 
@@ -171,7 +171,7 @@ function _edge(
         _variable_nodes(b, BT),
         _constraint_nodes(b, BT),
         node(b, set_objective_function_type(BT)),
-        cost(BT),
+        bridging_cost(BT),
     )
 end