Assuming that by "monoidal" you mean "strong monoidal", i.e. a functor $F$ comes with a natural *isomorphism* $f_{X,Y} : F(X\otimes Y) \cong F(X) \otimes F(Y)$ (and an isomorphism $i : F(1) \cong 1$, and these isomorphisms should intertwine the unit and associativity morphisms on the target category with the images under $F$ of the unit and associativity morphisms in the source), then every monoial functor is rigid, in the following sense:

Let $X$ have a left dual $X^*$, i.e. we have equipped the pair $(X,X^*)$ with a cap $X^* \otimes X \to 1$ and a cup $1 \to X\otimes X^*$, such that the zig-zag equations hold.
Then $F(X)$ has $F(X^*)$ as its left dual. Indeed, $i \circ F(\text{cap}) \circ f_{X^\ast,X}^{-1} : F(X^\ast) \otimes F(X) \cong F(X^\ast\otimes X) \to F(1) \cong 1$ and $f_{X,X^\ast} \circ F(\text{cup}) \circ i$ are a cap and cup between $F(X^*)$ and $F(X)$. (The zig-zag equations do require that the isomorphism $f,i$ intertwine the various associators and unitors.)

In particular, suppose that the source of your monoidal functor is rigid. Then the essential image of your functor lands within the largest rigid subcategory of the target. Moreover, suppose both source and target are rigid. The previous paragraph implies that there is a canonical (and unique) natural isomorphism $F(X^*) \cong F(X)^*$ intertwining the cups and caps.

It should be emphasized, however, that the axioms of rigid category do not require an isomoprhism between the left dual $X^\ast$ and the right dual $^\ast X$. (Now I wish I had chosen different left/right conventions. C'est la vie.) In many examples, these are isomorphic, and indeed in many examples one can choose a natural isomorphism $X^\ast \cong {^\ast X}$. For example, in any braided category, the braiding determines such an isomorphism. But actually the braiding determines many such isomorphisms, and there's no best way to choose one, and many errors have been made (e.g. by me) by pretending that $X^\ast = {^\ast X}$ in some canonical way. (In a symmetric monoidal category, there is a canonical isomorphism $X^\ast \cong {^\ast X}$. But it's not necessarily compatible with other structure you might have around.)

Long before thinking about quantum groups, a good example to work through is the category of supervector spaces (over a field in which $2$ is invertible). As a monoidal category, this is the category of $\mathbb Z/2$ modules, but the braiding is different. Namely, the braiding is uniquely determined (up to the usual words) by demanding that on the one-dimensional nontrivial representation $S$, the braiding $\beta_{S,S} : S\otimes S \to S\otimes S$ is multiplication by $-1$. There are isomorphisms $S\otimes S \cong 1$, with which you can present $S$ as its own dual. The canonical isomorphism (determined by the supervector space braiding) $S \cong ^\ast S \cong S^\ast \cong S$ with these most natural choices is then multiplication by $-1$, whereas the canonical isomorphism for the usual braiding is multiplicaiton by $+1$. Note that the identity monoidal functor from supervect to $(\mathbb Z/2)$-rep is not a braided functor.

The next example to think through is the "quantum group" related to the reductive algebraic group $\mathbb G_m$. As a monoidal category, this is the category of $\mathbb Z$-graded vector spaces. The braiding is determined by declaring that on the generating line $L$ (i.e. the one-dimensional vector space in grading $1$), we have $\beta_{L,L} : L\otimes L \to L\otimes L$ is multiplication by $q \in \mathbb K^\times$. This category has a skeletalization with one invertible object $[n] = L^{\otimes n}$ for each $n\in \mathbb Z$; general objects are formal direct sums of these. This skeletalization is a strict monoidal category (generically skeletalizations of monoidal categories have nontrivial associators). In this skeletalization, the braiding $\beta_{[m],[n]} : [m+n] = [m] \otimes [n] \overset\sim\to [n]\otimes[m] = [m+n]$ is multiplication by $q^{mn}$. This example shows all of the problems that come from being too cavalier about the difference between isomorphism and equality.

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