SOLVED IN COMMENTS
In my lectures, and these lecture notes on page 8 as an example, there is the relation
$$(((dF)X)f) \circ F = X(f \circ F)$$
where $F:M \rightarrow N$,
$dF:T_p(M) \rightarrow T_F(p)(N)$,
$X$ is a vector field on M and
$f\in C^\infty (N)$
The above are as claimed in the notes. Hoever it seems to me that, while these definitions work for the right hand side, they do not work for the left hand side. In particular, $f \circ F : M \rightarrow N$ but $X$ is a vector field on $M$.
I would appreciate if someone could clarify what is going on with the domains here.