Much of the theory of (weighted) (co)limits in a symmetric monoidal category $\mathcal{V}$ can be rephrased in terms of (co)ends. For example, the weighted colimit of a functor $F:\mathbf{C}\rightarrow\mathcal{V}$ with weight functor $W:\mathbf{C}^{op}\rightarrow\mathcal{V}$ can be written as the following coend: $$\mathrm{colim}^WF\cong\int^{c\in\mathbf{C}}Wc\otimes Fc.$$
Since $\mathcal{V}$ is symmetric, i.e. $v\otimes w\cong w\otimes v$ for all $v,w\in\mathrm{ob}(\mathcal{V})$, this seems to imply that this expression and, consequently, the weighted colimit is symmetric under the exchange of $F$ and $W$, is this correct?