At the end of section A.1.4 of the book "higher topos theory," there is a formula $X\otimes (C\otimes D)\simeq (X\otimes C)\otimes D$ which means the action property of tensoring in enriched categories. I suspect that $X\otimes (C\otimes D)\simeq (X\otimes D)\otimes C$ is right since, in this notation (i.e. $(A\otimes -) \dashv {}^A(-)$ in monoidal categories), the right adjoint of $(A\otimes B)\otimes -$ is ${}^{A\otimes B}(-) \cong ^{B}(^{A}(-))$.
So I think we should denote, in monoidal categories, the right adjoint of $A\otimes -$ (resp. $-\otimes A$) by $(-)^A$ (resp. $^A(-)$). But I found no one writing in this way (Joyal and Tierney are also using Lurie's order, using $(-)/A$ instead of $(-)^A$). I understand that it makes no difference when we are dealing with symmetric monoidal categories, but I wonder why everyone is using the opposite one. Is there any drawback using my notation?