Technically, it isn't an isomorphism (the word is abused very often), and there is no fixed, general syntactic correspondence. However, in the case of Lean, we

Technically, it isn't an isomorphism (the word is abused very often), and there is no fixed, general syntactic correspondence. However, in the case of Lean, we can establish a correspondence between its dependent type system and intuitionistic higher-order predicate logic.