We define the notion of a canonical envelope of a bilateral pairing and analyze canonical envelopes through bilateral density and compactness conditions. Canonical envelopes provide a systematic framework for a significant class of completion phenomena in mathematics, unifying these constructions through initial factorizations in categories of bilateral decompositions.The construction was motivated by Riehl's adjunction for weighted limits cite{riehl2008weighted}. We prove that all four Kan constructions (left and right Kan extensions and left and right Kan liftings) arise as instances of canonical envelopes (Corollary~ef{cor:kan-are-envelopes}), so weighted (co)limits and all four Kan constructions are recovered within the framework.We develop the theory of outer bilateral envelopes for cases where classical completions fail: the outer envelope objects $Y_Q$ and $X_Q$ always exist in presheaf categories, and the term emph{virtual canonical envelope} refers to this outer data when the interpolant is not yet known. We establish the geometric interpretation through cylinder factorization systems, connecting to Garner's work. We prove that canonical envelopes admit monadic organization: the canonical envelope functor, defined on the full subcategory of admissible pairings where CEs exist, extends to an idempotent monad whose Eilenberg-Moore algebras are precisely the complete bilateral pairings, with Garner's Isbell monad emerging as a natural specialization.We show that dicategories provide a bilateral algebraic presentation of dagger categories (Theorem~ef{thm:dicat-dagger-equiv}), with every dagger category admitting a canonical dicategory presentation and vice versa. The dicategory presentation, characterized as a canonical envelope (Theorem~ef{thm:dicat-envelope}), makes explicit the symmetric relationship between categorical and cocategorical composition that is implicit in the standard dagger category axiomatization. This bilateral presentation arises naturally from the canonical envelope construction and connects to Frobenius pseudomonoids, where the dicategory axioms correspond to Frobenius compatibility conditions.We construct canonical left and right envelope objects explicitly in presheaf categories via coend and end formulas, reducing the existence of a canonical envelope to a universal interpolation problem. We show that the interpolation problem is solvable in several classical settings, including: ind- and pro-completions cite{gabriel1971lokal}, Cauchy completions cite{lawvere1973metric}, Pratt's communes cite{pratt2010communes}, Isbell envelopes cite{isbell1960adequate}, and topological completions (Stone-Čech compactification, sobrification). We establish a structural correspondence with Pratt's commune theory for identity pairings, and compare structurally with Schoots's categorical canonical extensions cite{schoots2015generalising} and classical canonical extensions of distributive lattices cite{jonsson1951boolean,gehrke2001bounded}.