Digraph functors which admit both left and right adjoints

For our purposes, two functors Λ and Γ are said to be adjoint if for any digraphs G and  H, there exists a homomorphism of  Λ(G) to  H if and only if there exists a homomorphism of  G to  Γ(H). We investigate the right adjoints characterised by Pultr (1970). We find necessary conditions for these functors to admit right adjoints themselves. We give many examples where these necessary conditions are satisfied, and the right adjoint indeed exists. Finally, we discuss a connection between these right adjoints and homomorphism dualities.