def andy(a, b):
    inverse_a = {}
    for k,v in a.items():
        inverse_a[v.lower()] = str(k)
    #               ^              v   remove these lower() calls for case sensitivity
    return {k: [inverse_a.get(item.lower(), item) for item in v] for k,v in b.items()}

#Andy K's test case
a = {1: "Vue network", 2: "Blockbuster"}
b = {1: ['Mann theater','LA','vue network'], 2 : ['Apollo Theatre', 'NY', 'Blockbuster']}
expected_output = {1: ['Mann theater','LA','1'], 2 : ['Apollo Theatre', 'NY', '2']}
assert andy(a,b) == expected_output


#Kevin's test cases
#basic usage
assert andy(
    {1: "a", 2: "b"},
    {3: ["x", "y", "a"], 4: ["u", "v", "b"]}
    ) == {3: ["x", "y", "1"], 4: ["u", "v", "2"]}

#case insensitivity
assert andy(
    {1: "a", 2: "b"},
    {3: ["X", "Y", "A"], 4: ["U", "V", "B"]}
    ) == {3: ["X", "Y", "1"], 4: ["U", "V", "2"]}

#if b has no matches, return a plain copy of b
assert andy(
    {1: "foo", 2: "bar"},
    {3: ["x", "y", "a"], 4: ["u", "v", "b"]}
    ) == {3: ["x", "y", "a"], 4: ["u", "v", "b"]}

#if a is empty: same behavior as "b has no matches"
assert andy(
    {},
    {3: ["x", "y", "a"], 4: ["u", "v", "b"]}
    ) == {3: ["x", "y", "a"], 4: ["u", "v", "b"]}

#replacements can occur at any point in the list
assert andy(
    {1: "a", 2: "b"},
    {3: ["a", "y", "z"], 4: ["u", "b", "w"]}
    ) == {3: ["1", "y", "z"], 4: ["u", "2", "w"]}

#multiple replacements are possible
assert andy(
    {1: "a", 2: "b"},
    {3: ["a", "a", "a"], 4: ["a", "b", "a"]}
    ) == {3: ["1", "1", "1"], 4: ["1", "2", "1"]}

#if a has duplicate values, ignore all but the last one
assert andy(
    {1: "a", 2: "a"},
    {3: ["x", "y", "a"], 4: ["u", "v", "a"]}
    ) == {3: ["x", "y", "2"], 4: ["u", "v", "2"]}