  We recall several categories of graphs which are useful for describing
homotopy-coherent versions of generalized operads (e.g. cyclic operads, modular
operads, properads, and so on), and give new, uniform definitions for their
morphisms. This allows for straightforward comparisons, and we use this to show
that certain free-forgetful adjunctions between categories of generalized
operads can be realized at the level of presheaves. This includes adjunctions
between operads and cyclic operads, between dioperads and augmented cyclic
operads, and betweeen wheeled properads and modular operads.
