Let $S\subset M$ be an embedded submanifold. A vector field $X \in \mathfrak{X}(M)$ is said to be tangent to $S$ if $X_p \in T_pS$ for all $p \in S$. Show that for $Y \in \mathfrak{X}(M)$ tangent to $S$, there is a unique smooth vector field $X \in \mathfrak{X}(S)$ such that $X$ is $\iota$-related to $Y$.
I have seen that since each $Y_p \in \iota_{*p}(T_pS) \equiv T_pS$, we can assign tangent vectors $X_p \in T_pS$ pointwise, with $Y_p = \iota_{*p}X_p$. I'm having trouble showing that the resultant vector field $X$ is indeed smooth and unique.
One criterion is checking whether $Xf \in C^\infty(S)$ for all $f \in C^\infty(S)$. I am aware of the characterization $$ T_pS = \{X \in T_pM: Xf = 0 \;\text{ whenever } f \in C^\infty(M)\text{ and } f\big|_S = 0\}, $$ but I don't quite see if this helps.