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Let $M_1, M_2$ be smooth manifolds. $f: M_1 \times M_2 \longrightarrow M_1$ be the projection map.

Let $X: M_1 \longrightarrow TM_1$ be a vector field.

Show that there exists some vector field $$Y: M_1 \times M_2 \longrightarrow T(M_1 \times M_2)$$

Such that $Y \text{ and } X$ are $f$-related.

I need to verify that

$$(1) \ \ \ (f_*)_pY_p = X_{p_1} \forall p = (p_1,p_2)\in M_1 \times M_2 $$

which is equivalent to $\forall h \in C^{\infty}(M_1)$

$$(2) \ \ \ Y(h\circ f) = Xh \circ f $$

So I constructed such $Y$ where $Y_p = (X_{p_1}, 0_{p_2})$ where $0_{p_2} \in T_{p_2}M_2$ is just the zero derivation.

It seems "obvious" that the zero derivation doesn't do anything therefore the second condition is "automatic"? But I'm not sure how to formally state that and as of right now it seems sketchy.

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What you are doing is correct. I think what is confusing you is that you are implicitly using the isomorphism $T_{(p_1,p_2)}(M_1\times M_2)\cong T_{p_1}M_1\times T_{p_2}M_2$ when writing $Y_p=(X_{p_1},0_{p_2})$. But this isomorphism is given by the tangent maps of the the two projections. So defining $Y_p$ in the way you do actually says that $f_*(Y_p)=X_{f(p)}$ and thus the relatedness you want to prove.

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