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For our purposes, two functors {\Lambda} and {\Gamma} are said to berespectively left and right adjoints of each other if for any digraphs G and H,there exists a homomorphism of {\Lambda}(G) to H if and only if there exists ahomomorphism of G to {\Gamma}(H). We investigate the right adjointscharacterised by Pultr in [A. Pultr, The right adjoints into the categories ofrelational systems, in Reports of the Midwest Category Seminar, IV, volume 137of Lecture Notes in Mathematics, pages 100-113, Berlin, 1970]. We findnecessary conditions for these functors to admit right adjoints themselves. Wegive many examples where these necessary conditions are satisfied, and theright adjoint indeed exists. Finally, we discuss a connection between theseright adjoints and homomorphism dualities.

Digraph functors which admit both left and right adjoints