Duals as right adjoints

This is extremely incomplete and preliminary.

$$\boxed { \begin{array} {}  &&&& \leftcat 1  &&&&  {\rightadj t}_{\rightcat{C^n}}     \\  & \leftadj v && \rightadj{b^i}  && \leftadj{b_i} &&&   \\  \hline   &&  \leftcat{  k^1 \rlap{ \xleftarrow [\kern 10em] {\displaystyle 1\in k^1} }  } &&&& \leftcat{ k^1} \rightadj{   \rlap{  \xleftarrow [\kern 10em] { \displaystyle t_{\rightcat{C^n}} \in [\leftadj{B^m},\rightcat{C^n}]  }  }   } &&&& \leftcat{ k^1 }   \\     & \leftadj{ \swarrow  \rlap{\kern-4em v\in B^m}  }  &  \rightadj{ \big\Downarrow \rlap{\epsilon_{\rightcat{k^1}}}  }  &   \rightadj{ \nwarrow \rlap{ \kern-.5em B^* } } & \leftadj{ \big\Downarrow \rlap\eta }  & \leftadj{ \swarrow  \rlap{\kern-.5em B^m}  }  &  \rightadj{ \big\Downarrow \rlap{\epsilon_{\rightcat{k^1}}}  }  &   \rightadj{ \nwarrow \rlap{ \kern-.5em B^* } }   &  \rightcat{ \swarrow \rlap{\kern-.5em C^n} }    &   \rightcat{ \swarrow \rlap{C^n} }   \\   \rightcat{ k^1 \rlap{\xleftarrow {\kern 10em} }  } &&&& \rightcat{ k^1 \rlap{\xleftarrow {\kern 10em} }  } &&&& \rightcat{ k^1 }    \end{array} }$$

$$\boxed { \begin{array} {} && \leftadj v \tensor {\rightadj t}_{\rightcat{C^n}}  \\  && \Big\downarrow  \\  &&  \leftadj{ v \tensor {} } \leftcat 1 \rightadj{ {} \tensor t_{\rightcat{C^n}} }  \\  &&  \leftadj{     \llap{   B^m \tensor \eta  \rightadj{ {} \tensor [\leftadj{B^m}, \rightcat{C^n}] }   }   \Big\downarrow   }  &    \leftcat{  \searrow \rlap{ \leftadj v \tensor \zeta\inv }  }   \\    && \leftadj v \tensor ( \Sigma \rightadj{b^i} \tensor \leftadj{b_i} ) \tensor {\rightadj t}_{\rightcat{C^n}} &&    \\   & \rightadj{  \llap{ \epsilon_{\rightcat{k^1}} \tensor \leftadj{B^m} \tensor [\leftadj{B^m}, \rightcat{C^n}]  } \swarrow   } &&  \rightadj{  \searrow \rlap{ \leftadj v \tensor ( \black\Sigma b^i  \tensor \epsilon_{\rightcat{C^n}} ) }  } &   \\     \leftadj{   \big( \Sigma \rightadj{ \langle \leftadj v, b^i \rangle }  \tensor \leftadj{b_i} \big)  } \rightadj{ {} \tensor t_{\rightcat{ C^n}}  }   && \rightadj{  \epsilon_{\rightcat{k^1}} \tensor \epsilon_{ \rightcat{C^n} }  }     && \leftadj v \tensor \big( \Sigma \rightadj{b^i} \tensor  \rightadj { \langle \leftadj {b_i}, t_{\rightcat{C^n} }  \rangle } \big)   \\    \Vert & \rightadj{  \llap{ \rightcat{k^1} \tensor \epsilon_{\rightcat{C^n}}  } \searrow   }  && \rightadj{  \swarrow \rlap{ \epsilon_{\rightcat{k^1}} \tensor \rightcat{k^1} }  } &  \\   \leftadj{   \rightcat 1 \tensor \Sigma  \rightadj{ \langle \leftadj v, b^i \rangle } \leftadj{b_i}  } \rightadj{ {} \tensor t_{\rightcat{ C^n}}  }  && \Sigma \rightadj{ \langle \leftadj v, b^i \rangle } \tensor \rightadj{ \langle \leftadj b_i , t_{\rightcat{C^n}}  \rangle }   \\  \Vert  && \Big\Vert \rlap{   \leftadj{ \tensor \text{b.l.} } , \rightadj{  \langle,\rangle \text{ homog. and additive (thus b.l.)}  }   }  \\  \leftadj{   \rightcat 1 \tensor v  } \rightadj{ {} \tensor t_{\rightcat{ C^n}}  }   && \rightcat 1 \tensor \rightadj{ \big\langle \black{\displaystyle\Sigma} \langle \leftadj v, b^i \rangle  \leftadj b_i , t_{\rightcat{C^n}}  \big\rangle }   \\   &  \rightadj{  \llap{ \rightcat{k^1} \tensor \epsilon_{\rightcat{C^n}}  } \searrow   }   & \Big\Vert   \\ && \rightcat 1 \tensor \rightadj{ \big\langle \leftadj v, t_{\rightcat{C^n}}  \big\rangle  }   \\    \end{array} }$$