Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) to $\mathbb{R}$ and let $F_T$ denote the pullback of $T$, i.e. the map $F_T: \mathcal{F}(N,\mathbb{R}) \to \mathcal{F}(M,\mathbb{R})$ defined by $F_T(f)= f \circ T$.

Given an arbitrary map $F: \mathcal{F}(N,\mathbb{R}) \to \mathcal{F}(M,\mathbb{R})$, is there a way to determine if it is of the form $F_T$ for some $T$, or to associate with it a closest map $F_T$ of this form (i.e. with respect to some metric) and to determine the corresponding map $T$? Also, could there be some tweaking to the problem statement here that would make this possible?

Thank you very much.