Modern software is no more developed in a single programming language. Instead, programmers tend to exploit cross-language interoperability mechanisms to combine code stemming from different languages, and thus yielding fully-fledged multi-language programs. Whilst this approach enables developers to benefit from the strengths of each single-language, on the other hand it complicates the semantics of such programs. Indeed, the resulting multi-language does not meet any of the semantics of the combined languages. In this paper, we broaden the boundary functions-based approach à la Matthews and Findler to propose an algebraic framework that provides a constructive mathematical notion of multi-language able to determine its semantics. The aim of this work is to overcome the lack of a formal method (resp., model) to design (resp., represent) a multi-language, regardless of the inherent nature of the underlying languages. We show that our construction ensures the uniqueness of the semantic function (i.e., the multi-language semantics induced by the combined languages) by proving the initiality of the term model (i.e., the abstract syntax of the multi-language) in its category.