I'm having a bit of difficulty understanding $F$-related vector fields. I think I understand conceptually what's going on (taking a vector field on a manifold $M$ and getting a smooth vector field on $N$ by applying the differential of a map), but I'm having trouble applying the concept.
For example, here's a problem (8.14 in Lee's Smooth Manifolds textbook) Let $M,N$ be smooth manifolds with or without boundary and let $f:M\to N$ be a smooth map. Define $F:M\to M\times N$ by $F(x)=(x,f(x))$. Show that for each vector field $X$ on $M$, there is a smooth vector field $Y$ on $M\times N$ which is $F$ related to $X$.
Now, I see that $F$ is the graph of $f$, and so $F(M)$ is closed in $M\times N$, and there's a proposition in Lee which says that if we have a smooth vector field on a closed subset, we can extend it to a smooth vector field on the whole manifold. However, I'm really not sure where to go. I see that $F(M)\subset M\times N$ is an embedded manifold, so we have slice charts, but I'm struggling to see how to construct this $Y$.