1. Suppose \$$\def\cat#1{{\mathcal{#1}}}\$$ \$$\def\comp#1{{\widehat{#1}}}\$$ \$$\def\conj#1{{\check{#1}}}\$$ \$$\def\id{{\mathrm{id}}}\comp{F}=\conj{G}\$$. By the definitions of companion and conjoint, \$$\cat{Q}(F(p),q)=\comp{F}(p,q)=\conj{G}(p,q)=\cat{P}(p,G(q))\$$ for every \$$p\in\cat{P}\$$ and \$$q\in\cat{Q}\$$. Suppose \$$F\$$ and \$$G\$$ are \$$\cat{V}\$$-adjoints. Since \$$\cat{V}\$$ is a skeletal quantale, \eqref{eq1} holds with equality. By the definitions of companion and conjoint \$$\conj{G}(p,q)=\cat{P}(p,G(q))=\cat{Q}(F(p),q)=\comp{F}(p,q)\$$ holds for every \$$p\in\cat{P}\$$ and \$$q\in\cat{Q}\$$. Thus, \$$\comp{F}=\conj{G}\$$.

2. \$$\cat{P}(p,\id(q))=\cat{P}(p,q)=\cat{P}(\id(p),q)\$$, so \$$\comp{\id}=\conj{\id}\$$.