We present the new combinatorial class of product-coproduct prographs which are planar assemblies of two types of operators: products having two inputs and a single output and coproducts having a single input and two outputs. We show that such graphs are enumerated by the $3$-dimensional Catalan numbers. We present some combinatorial bijections positioning product-coproduct prographs as key objects to probe families of objects enumerated by the $3$-dimensional Catalan numbers.