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 in [A. Pultr, The right adjoints into the categories of relational systems, In Reports of the Midwest Category Seminar, IV, volume 137 of Lecture Notes in Mathematics, pages 100–113, Berlin, 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. AMS subject classification: 05C20, 18A40, 05C60