  This paper describes several cases of adjunction in the homomorphism preorder
of relational structures. We say that two functors $\Lambda$ and $\Gamma$
between thin categories of relational structures are adjoint if for all
structures $\mathbf A$ and $\mathbf B$, we have that $\Lambda(\mathbf A)$ maps
homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically
to $\Gamma(\mathbf B)$. If this is the case, $\Lambda$ is called the left
adjoint to $\Gamma$ and $\Gamma$ the right adjoint to $\Lambda$. In 2015,
Foniok and Tardif described some functors on the category of digraphs that
allow both left and right adjoints. The main contribution of Foniok and Tardif
is a construction of right adjoints to some of the functors identified as right
adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to
arbitrary relational structures, and coincidently, we also provide more right
adjoints on digraphs, and since these constructions are connected to finite
duality, we also provide a new construction of duals to trees. Our results are
inspired by an application in promise constraint satisfaction -- it has been
shown that such functors can be used as efficient reductions between these
problems.
