Clonoids of Boolean functions with a monotone or discriminator source clone

Extending Sparks's theorem, we determine the cardinality of the lattice of (C₁,C₂)-clonoids of Boolean functions in the cases where the target clone C₂ is the clone of projections. Moreover, we explicitly describe the (C₁,C₂)-clonoids of Boolean functions in the cases where the source clone C₁ is one of the four clones of monotone functions or contains the discriminator function.