  We generalize the correspondence between theories and monads with arities of
arXiv:1101.3064 to $\infty$-categories. Additionally, we introduce the notion
of complete theories that is unique to the $\infty$-categorical case and
provide a completion construction for a certain class of theories. Along the
way we also develop the necessary technical material related to the flagged
bicategory of correspondences and lax functor in the $\infty$-categorical
context.
