Assume $M_1, \dots,M_k$ are smooth manifolds and define $M:=M_1\times \dots \times M_k$. Denote the projections on the $i$-th factor with $\pi_i: M \rightarrow M_i$. I want to show that for each smooth vector field $X_i \in \Gamma(M_i,TM_i)$ there is a $\pi_i$-related smooth vector field $Y\in \Gamma(M,TM)$.
Since I don't know any theorems about the existence for related vector fields my approach was to prove the existence by constructing one. I know that if $Y$ is a smooth vector field over $M$ then for all smooth function $f\in C^\infty(M)$, $fY:M\rightarrow TM$, defined by $$(fY)_p=f(p)Y_p$$ is a smooth vector field as well.
From the Lemma below I know that for each real-valued smooth function $g$ on an open subset of $M_i$, we have $$Y(g\circ \pi_i)=(Xg)\circ \pi_i.$$
Well, that's basically how far I am right. I have read the chapter about this topic in Introduction to smooth manifolds by John M. Lee but I am still lacking intuition for this situation. If anyone could lead me in the right direction I would appreciate it.
Definition of $F$-related vector fields:
Suppose $F: M\rightarrow N$ is a smooth, where $M,N$ are smooth manifolds. Smooth vector fields $X\in \Gamma(M,TM)$ and $Y\in \Gamma(N,TN)$ are called $\mathbf{F}$-related, if for each $p\in M$, $dF_p(X_p)=Y_{F(p)}$.
Lemma:Assume $X,Y$ and $F$ are as specified in the definition above. $X$ and $Y$ are $F$-related if and only if for every smooth real-valued function $f$ on an open subset $U\subseteq N$ we have $X(f\circ F)=(Yf)\circ F$. This Lemma follows basically by inserting in the definitions.