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