I was reading this definition from journal article 'fixed-point logics with nondeterministic choice' by Anuj Dawar and David Richerby. On page 505 it says
'Classes of structures are assumed to be isomorphism-closed: if a structure is in a class, all images of that structure under isomorphisms are in the class. If $\mathcal{C}$ is a class of structures, a k-ary query on $\mathcal{C}$ maps each structure $\mathcal{U}\in\mathcal{C}$ to a k-ary relation on |$\mathcal{U}$| such that if $\rho$ : $\mathcal{U}\rightarrow\mathcal{B}$ is an isomorphism, $\mathcal{Q}(\mathcal{B})$ = $\rho(\mathcal{Q}(\mathcal{U}))$'.
I am trying to understand how we have an isomorphism between structures by some example? Also what exactly query is needed for? Why do we have need to map a structure to a k-ary relation on its universe?