Let $(M,\otimes)$ be a rigid monoidal category, for which left and right duals coincide. For any object $X \in M$, we can define a monoid structure on $X \otimes X^*$: Multiplication is defined by evaluation $ev$ as follows: $$ id \otimes ev \otimes id: X \otimes X^* \otimes X \otimes X^* \to X \otimes X^*. $$ The unit is just given by coevaluation in the obvious way.

Is this definition correct? I can't seem to convince myself that the associativity and unit axioms of a monoid object are true.

Let $(M,\otimes)$ be a rigid monoidal category, for which left and right duals coincide. For any object $X \in M$, we can define a monoid structure on $X \otimes X^*$: Multiplication is defined by evaluation $ev$ as follows: $$ id \otimes ev \otimes id: X \otimes X^* \otimes X \otimes X^* \to X \otimes X^*. $$ The unit is just given by coevaluation in the obvious way.

Is this definition correct? I can convince myself that the unit axiom of a monoid object is true - it follows from the axioms of a dual. But I'm confused about the associativity axiom.