Adjoint functor theorems for homotopically enriched categories

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal{V}$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched version of Freyd’s adjoint functor theorem. For non-trivial model structures, we obtain new adjoint functor theorems of a homotopical flavour - in particular, when $\mathcal{V}$ is the category of simplical sets we obtain a homotopical adjoint functor theorem appropriate to the $\mathrm{\infty }$-cosmoi of Riehl and Verity. We also investigate accessibility in the enriched setting, in particular obtaining homotopical cocompleteness results for accessible $\mathrm{\infty }$-cosmoi.