This is the 3180th most frequent Russian word.
московскому
"To the Moscow (adj.)" or "Moscow's (adj. dative singular form)."
The word 'московскому' is used here in the dative case to indicate the indirect object being the museum located in Moscow.
Я подарил книгу московскому музею.
I gifted a book to the Moscow museum.
Here, 'московскому' is in the dative case to express what the action of dedication is directed towards.
Он посвящает своё время московскому культурному наследию.
He dedicates his time to the Moscow cultural heritage.
In this example, 'московскому' is also in the dative case, indicating the recipient of the questions.