By definition of the identity \\(\newcommand{\idprof}[1]{\mathrm{id}(#1)}\newcommand{\cat}[1]{\mathcal{#1}}\newcommand{\companion}[1]{\widehat{#1}}\cat{V}-\\)functor, \\(\cat{P}(\idprof{p},\idprof{q})=\cat{P}(p,q)\\). By the definitions of companion (, conjoint) and unit profunctor, \\(\companion{\mathrm{id}}(p,q)=\cat{P}(\idprof{p},q)=\cat{P}(p,q)=\cat{P}(p,\idprof{q})=\check{\mathrm{id}}(p,q)=U\_\cat{P}(p,q)\\).