We apply the theory of binding algebra to syntax with linear binders. We construct a category of models for a linear binding signature. The initial model serves as abstract syntax for the signature. Moreover it contains structure for modelling simultaneous substitution. We use the presheaf category on the free strict symmetric monoidal category on 1 to construct models of each binding signature. This presheaf category has two monoidal structures, one of which models pairing of terms and the other simultaneous substitution.