Prove that there is unique simplician map $f\colon \sigma \to \tau$, whose restiction if $f_{0}$.

Question

In Lee's book ''Introduction to Topological manifolds'' we have the following definition :
Def: A map $f\colon \sigma\to \tau$ between two simplices is call simplicial map if it is the restriction of an affine map that takes vertices of $\sigma$ to vertices of $\tau$.
We want to prove that given any map $f_{0}$ from the set of vertices of $\sigma$ to the set of vertices of $\tau$ there is unique simplicial map $f\colon \sigma\to \tau$ whose restriction to the vertices of $\sigma$ is $f_{0}$.
I have two questions :
(a) What is the importance of affine extension of f, on the definition ?
(b) My idea of the claim is the following :
If $V_{1}= \\{ v_{1},\dotsc,v_{k} \\},\ V_{2}=\\{ u_{1},\dotsc,u_{m} \\}$ the sets of vertices of $\sigma$ and $\tau$ resp. and $f_{0}(v_{j})=u_{i_{j}}$ we consider the extension $$f\left(\sum_{j=1}^{k}t_{j}v_{j}\right)=\sum_{j=1}^{k}t_{j}u_{i_{j}},\quad \sum_{j}t_{j}=1.$$ Can we find a affine map such that its restriction at $\sigma$ is equal to f ?