Matrices

$$\boxed{ \begin{array} {} 1 \\ & \leftcat\searrow &&& \leftcat\searrow \rlap{ \hom {\rightcat -} M {\leftcat i} } \\ && \leftcat{A_i} \rlap{   \leftcat{  \xrightarrow [\kern12em] { \textstyle \text{column vectors} \atop \textstyle \text{vectors} }  }   } &&&& \rightcat{ \displaystyle \mathop{\rightadj\prod}_{j=1} ^n B_j }  \\ & \leftcat{ \llap{x_i} \nearrow } && \leftadj{   \searrow \rlap{ \kern-2em \iota_{\rightcat i} = \widehat {e_{\leftcat i}}  }    }  && \searrow \rlap{ \hom {\rightcat j} M {\leftcat i} } && \rightadj{ \searrow \rlap{ \pi_{\rightcat j} } } \\ 1 \rlap{ \leftcat{ \xrightarrow[ \textstyle \text{either $x_ie_i$ or $X$} ] {\kern10em} } }  &&&& \leftcat{ {\displaystyle \mathop{\leftadj\sum}_{i=1} ^m} A_i } \rlap{ \rightcat{  \xrightarrow [ \textstyle \text{row vectors} \atop \textstyle \text{covectors} ] {\kern12em}  }   } &&&& \rightcat{B_j} \\ \\  \end{array} }$$