If $X, Y$ are objects in monoidal category $\mathcal{C}$ which have left duals $X^∗, Y^∗$ and $f : X → Y $ is a morphism in $\mathcal{C}$, then the left dual map $f^∗ : Y^∗ → X^∗$ of $f$ is given by:
$f^* = (ev_Y ⊗id_{X^*})((id_{Y^*} ⊗ f )⊗id_{X^*})(id_{Y^*} ⊗coev_X)$
where $ev_X: X^∗⊗X → 1$, $coev_X : 1 → X ⊗X^∗$, and $ev_Y: Y^∗⊗Y → 1$, $coev_Y : 1 → Y ⊗Y^∗$, are the evaluation and coevaluation corresponding to $X,Y$ respectively. I am trying to show that if objects in $X,Y,Z \in \mathcal{C}$ which have left duals $X^*,Y^*,Z^*$, and $f : X → Y, g : Y → Z$ are morphisms in $\mathcal{C}$, then $(gf)^*=f^*g^*$. Is the definition of of $f^*$ is equivalent to the following:
A left dual map of $f$ is the morphism $f^∗ : Y^∗ → X^∗$ which is uniquely determined by the commutative diagram
