I was readying the construction of the Tensor Product that was made on the book Introduction to Smooth Manifolds, from John M. Lee. In the proposition 12.7 he proved that his construction satisfied the universal property of the Tensor Product. His proof follows:
where $\mathfrak{F}(V\times W)$ denotes the free vector space over $V\times W$, and $\mathcal{R}$ is the subspace spanned by the vector of the form $$ \begin{array}{c} (v + v',w) - (v,w) - (v',w)\\ (v, w + w') - (v,w) - (v,w')\\ (\lambda v,w) - \lambda(v,w)\\ (v,\lambda w) - \lambda(v,w)\\ \end{array} $$ and $\Pi$ is the natural projection that sends each element of $\mathfrak{F}(V\times W)$ on it's equivalence class in $\mathfrak{F}(V\times W)/\mathcal{R}$.
My doubt about his proof is the step: $\mathcal{R}\text{ contained in }ker(\overline{A}) \Rightarrow \overline{A}\text{ descends to a linear map $\widetilde{A}$ such that} \widetilde{A}\circ\Pi = \overline{A}$. Where did he get this implication from? and why $\widetilde{A}$ is a linear map, exactly?
I'm sorry if it's too much text and sorry about the image, but I'm really confused about this step.
Your doubt is a well know theorem in linear algebra. You can find it in any linear algebra text worth reading or for example here on the second page.
The basic idea is to define $\tilde{A} ([x]) = \overline{A} (x)$, where $[x] = \Pi (x)$ is the equivalence class of $x \in \mathfrak{F}(V \times W)$. This map ($\tilde{A}$) is well defined, because $\mathcal{R} \subset \ker \overline{A}$ and linear, because $\overline{A}$ is.