If we use the algebraic definition of a vector field, then a vector field $Y$ on $N$ is $F$-related to a vector field $X$ on $M$ iff for all $g\in C^\infty(N)$: $X(g\circ F)=(Yg)\circ F$.

Suppose $X=\sum\z_j\pa_{x_j}$, $Y=\sum\eta_k\pa_{y_k}$, $x\in M$ and $y=F(x)$, then by the chain rule $$ X(g\circ F)(x) =\sum_{j,k}\z_j(x)\pa_{y_k}g(y)\pa_jF_k(x) =\sum_kdF_k(x)(\z(x))\pa_{y_k}g(y) $$ and $$ (Yg)\circ F(x) =\sum_k\eta_k(y)\pa_{y_k}g(y)~. $$ Thus the relation $X(g\circ F)(x)=(Yg)\circ F(x)$ for all $g\in C^\infty(N)$ is equivalent to: for all $k$: $F_k(x)(\z(x))=\eta_k$, i.e. $DF(x)(\z(x))=\eta(F(x))$.