Let $\mathbb{F}=\{f^{(s)}\}_{s\in{\mathbb{R}^d}}$ a collection of tempered distributions in $\mathbb{R}^d$ and $F=\{h\in S(\mathbb{R}^d)\,:\,\mbox{the map }s\mapsto f^{(s)}(h)\mbox{ is in }S(\mathbb{R}^d)\}$.
Is there some class of general conditions about $\mathbb{F}$ in order to obtain $\boldsymbol{\langle f^{(\cdot)}(u),v\rangle_{L^2(\mathbb{R}^d)}=\langle u,f^{(\cdot)}(v)\rangle_{L^2(\mathbb{R}^d)}}$ for all $\boldsymbol{u,v\in F}$?
That is: Are there some set of conditions over $\mathbb{F}$, say $P(\mathbb{F})$, such that bold text is true if and only if $P(\mathbb{F})$ is true.
Note: The inner product is explicitelly $$\int_{\mathbb{R}^d}f^{(s)}(u)v(s)\,\mathrm{d}s=\int_{\mathbb{R}^d}u(s)f^{(s)}(v)\,\mathrm{d}s.$$