Full Idea
Tarski's account doesn't hold for natural languages. The general notion of truth is replaced by "true-in-L", where L is a formal language. Hence truth is relativized to each artificial language.
Gist of Idea
Tarski made truth relative, by only defining truth within some given artificial language
Source
report of Alfred Tarski (The Concept of Truth for Formalized Languages [1933]) by Paul O'Grady - Relativism Ch.2
Book Reference
O'Grady,Paul: 'Relativism' [Acumen 2002], p.32
A Reaction
This is a pretty good indication that Tarski's theory is NOT a correspondence theory, even if its structure may sometimes give that impression.