Clonoids of Boolean functions with a monotone or discriminator source clone

Extending Sparks’s theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions in the cases where the target clone $C_2$ is the clone of projections. Moreover, we explicitly describe the $(C_1,C_2)$-clonoids of Boolean functions in the cases where the source clone $C_1$ is one of the four clones of monotone functions or contains the discriminator function.