  Given a class $\mathcal C$ of models, a binary relation ${\mathcal R}$
between models, and a model-theoretic language $L$, we consider the modal logic
and the modal algebra of the theory of $\mathcal C$ in $L$ where the modal
operator is interpreted via $\mathcal R$. We discuss how modal theories of
$\mathcal C$ and ${\mathcal R}$ depend on the model-theoretic language, their
Kripke completeness, and expressibility of the modality inside $L$. We
calculate such theories for the submodel and the quotient relations. We prove a
downward L\"owenheim--Skolem theorem for first-order language expanded with the
modal operator for the extension relation between models.
