How to make a composite function out of 2 vector fields?

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Ok, so currently I'm studying differential calculus on vector fields, but I feel like I'm missing some understanding on this specific topic: how do I make a composite function out of 2 vector fields ? for example:

being $f(x,y,z) = (x^2+y+z)i + (xy)j $

and $g(u,v,w) = (u+v+w)i + (w+v)j+(u*\sin(v))k $

find $h(u,v,w)$ such as:

$h(u,v,w) = f(g(u,v,w))$

OBS: That is just a simple example, please don't focus on it's answer. I'm really interested in how it is done