  Eight categorical soundness and completeness theorems are established within
the framework of algebraic theories. Exactly six of the eight deduction systems
exhibit complete semantics within the cartesian monoidal category of sets. The
multicategorical meta-theorem via soundness and completeness enables the
transference of properties of families of models from the cartesian monoidal
category of sets to $\Delta$-multicategories $C$.
  A bijective correspondence $R \mapsto \Delta_R$ is made between context
structures $R$ and structure categories $\Delta$, which are wide subcategories
of $\textbf{FinOrd}$ consisting of finite ordinals and functions. Given a
multisorted signature $\sigma$ with a context structure $R$, an equational
deduction system $\vdash_R$ is constructed for $R$-theories. The models within
$\Delta_R$-multicategories provide a natural semantic framework for the
deduction system $\vdash_R$ for modelable context structures $R$. Each of the
eight modelable context structures $R$ is linked with a soundness and
completeness theorem for the deduction system $\vdash_R$.
