We present a systematic algebraic approach to the weak coupling of Cauchy problems for multiple Lohe tensor (LT) models. To this end, we associate each admissible Cauchy problem for the LT model with a characteristic symbol, which is a 4-tuple consisting of a size vector, a natural frequency tensor, a coupling strength tensor, and an admissible initial configuration. In this way, the collection of all admissible Cauchy problems for LT models is in one-to-one correspondence with the space of characteristic symbols. We then introduce a binary operation called the fusion operation, which acts between characteristic symbols. This operation is associative and admits an identity element, thereby making the space of characteristic symbols a monoid. Using the fusion operation, a weakly coupled system of multiple LT models can be constructed by applying it to the corresponding characteristic symbols. As a concrete example, we consider the weak coupling of the swarm sphere model and the Lohe matrix model, and provide a sufficient framework under which the proposed weakly coupled system exhibits emergent dynamics.