\\(\def\cat#1{{\mathcal{#1}}}\\) \\(\def\comp#1{{\widehat{#1}}}\\) \\(\def\conj#1{{\check{#1}}}\\) \\(\def\id{{\mathrm{id}}}\\)

Let \\(\cat{V}\\) be a skeletal quantale, let \\(\cat{P}\\) and \\(\cat{Q}\\) be \\(\cat{V}\\)-categories, and let \\(F\colon\cat{P}\to\cat{Q}\\) and \\(G\colon\cat{Q}\to\cat{P}\\) be \\(\cat{V}\\)-functors.

1. Show that \\(F\\) and \\(G\\) are \\(\cat{V}\\)-adjoints (as in \ref{eq1}) if and only if the companion of the former equals the conjoint of the latter: \\(\comp{F}=\conj{G}\\).

2. Use this to prove that \\(\comp{\id}=\conj{\id}\\), as was stated in 4.34.

\begin{equation}\label{eq1}\tag{4.39}\cat{P}(p,G(q))\cong\cat{Q}(F(p),q)\end{equation}

Let \\(\cat{V}\\) be a skeletal quantale, let \\(\cat{P}\\) and \\(\cat{Q}\\) be \\(\cat{V}\\)-categories, and let \\(F\colon\cat{P}\to\cat{Q}\\) and \\(G\colon\cat{Q}\to\cat{P}\\) be \\(\cat{V}\\)-functors.

1. Show that \\(F\\) and \\(G\\) are \\(\cat{V}\\)-adjoints (as in \ref{eq1}) if and only if the companion of the former equals the conjoint of the latter: \\(\comp{F}=\conj{G}\\).

2. Use this to prove that \\(\comp{\id}=\conj{\id}\\), as was stated in 4.34.

\begin{equation}\label{eq1}\tag{4.39}\cat{P}(p,G(q))\cong\cat{Q}(F(p),q)\end{equation}