A^l\to A^l\otimes I\xrightarrow{\eta^l}A^l\otimes (A\otimes A^l)\to (A^l\otimes A)\otimes A^l \xrightarrow{\epsilon^l} I\otimes A^l\to A^l .
Non-symmetric compact closed categories find applications in linguistics, in the area of categorial grammars and specifically in pregroup grammars, where the distinct left and right adjoints are required to capture word-order in sentences.