What does "factors through" mean in $K[K^t] \xrightarrow{\phi} K[K^n] \xrightarrow{\rho} K[V]$ factors through $K[W]$ $(*)$?
$V \subseteq K^n, W \subseteq K^t$ are varieties, $K[V] = K[x_1, \dots, x_n]/I(V)$ is the coordinate ring of $V$ and similarly for $K[W], \rho$ is the restriction map, $\Phi : K^n \to K^t$ is a polynomial map, and $\phi : K[K^t] \to K[K^n]$ sends $f \to f \circ \Phi.$
The problem is to prove $(*)$ is equivalent to $\Phi$ restricting to a map $\Phi|_V \to W,$ which is equivalent to $\Phi(V) \subseteq W.$ As the previous part of the problem asked to prove $\Phi(V) \subseteq W \Leftrightarrow \phi(I(W)) \subseteq I(V),$ I suspect $(*)$ is equivalent to $\phi(I(W)) \subseteq I(V),$ which is equivalent to $\phi(I(W)) \subseteq \ker \rho.$ This is starting to look like exact sequences, but I don't want to waste any time under the assumption we can work backwards to guess what "factors through" means.
As noted in the comments by FShrike, saying a map $f:X\to Y$ "factors through $Z$" means that there are maps $g:X\to Z$ and $h:Z\to Y$ such that $f=h\circ g$. In your case, you're being asked to show that the composite map $K[K^t]\to K[K^n]\to K[V]$ can also be written $K[K^t]\to K[W]\to K[V]$.