A properad in a symmetric monoidal category$C$ is a monoid in the monoidal category of bisymmetric sequences in $C$ (i.e., functors $\Sigma\times\Sigma\to C$) equipped with a version of substitution product modeled on connected directed graphs with 2 levels instead of corollas, which are used for operads. See §1.2 in Vallette for details. The idea is that operations in a properad can have multiple inputs and outputs, as opposed to a single output in an operad.

The pluricategories of Kavanagh (Definition 2.1.11) are similar to properads, except that they have identity morphisms for each list of objects, rather than a unary morphism for each object.