On rectification and enrichment of infinity properads

We develop a theory of infinity properads enriched in general symmetric monoidal category. These are defined as presheaves, satisfying Segal condition and Rezk completeness condition, over certain categories graphs. In particular, we introduce new category level graphs which also allow us to give framework for algebras an properad. show that one can vary the without changing underlying theory. cannot always be rectified, indicating conjecture second author Robertson is unlikely hold. This stands stark contrast situation operads, further demarcate these situations by examining cases dioperads output properads. both cases, provide rectification theorem says each up-to-homotopy object equivalent strict one.