Different amounts of abstraction lend themselves well to problem solving at different "zoom-levels". Discrete finite quantum spaces, also known as (finite) quantum sets, can be abstracted in many ways. Quantum sets are represented as tuples of Hilbert spaces, as multi-matrix algebras (finite-dimensional C$^*$ algebras), as special symmetric dagger Frobenius algebras, or as wires in string diagrams. They can be more tersely described by tuples of natural numbers or by partitions of finite sets. We connect these different levels of abstraction. In particular, we present insights of the string diagrams perspective for the following ideas: complete positivity, measurement, quantum graphs, quantum teleportation, and quantum games.