We provide a framework to deal with "diagrammatic" operadic actions in Cat,
i.e. actions given by compositions of diagrams, rather than strings of objects.
We achieve this by introducing a monoidal structure on the category of small
diagrams in Cat, which generalizes simultaneously the composition product of
collections in the theory of operads, and the semi-direct product of groups.
Familial operads are given then as monoids with respect to this monoidal
structure, and algebras are defined as categories, carrying actions of such
monoids.