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}\\).