Enriched adjoint functor theorem for cotensored categories

Assume first that $F{⊣}_{\mathcal{V}}G$ is an enriched adjunction. Applying the hom-object isomorphism of [L1] to the cotensor object $X⋔B$ and then reading the cotensor universal properties from [L2] shows that $G(X⋔B)$ represents the same functor as $X⋔GB$. Therefore $G$ preserves cotensors.

Conversely, assume $G$ has the stated $\mathcal{V}$-functor structure and coherent cotensor-preservation isomorphisms. For each base object $X$, the ordinary adjunction identifies maps $FA\to X⋔B$ with maps $A\to G(X⋔B)$. Cotensor preservation rewrites the target as $A\to X⋔GB$, and applying the cotensor representing property of [L2] again converts this into maps $X\to \mathcal{A}(A,GB)$ naturally in $X$. By Yoneda in the base, these natural bijections determine a $\mathcal{V}$-natural isomorphism $\mathcal{B}(FA,B)\cong \mathcal{A}(A,GB)$, giving [L1] and the compatible enriched structure on $F$.

Thus a coherent enriched structure on the right adjoint together with cotensor preservation is equivalent to lifting the underlying adjunction to an enriched one. The tensor statement is dual.