The tangent space of the affine Minkowski model at any event is canonically V

Fix $v\in V$ and a smooth germ $[f]$ at $p$. By [L6] the curve $c(t):=p+tv$ has coordinate representation $x(c(t))=x(p)+tu$ with $u=E^{-1}(v)$, a smooth curve in ${\mathbb{R}}^4$; hence $c$ is a smooth curve through $p$ in the chart $x$, and $t\mapsto f(c(t))$ is smooth near $0$ by [L2], so its derivative at $0$ exists. If $g$ agrees with $f$ near $p$, then $f\circ c$ and $g\circ c$ agree near $0$, so the derivative depends only on the germ $[f]$; hence ${\mathrm{\Phi }}_p(v)([f])$ is a well-defined real number.

With $u=E^{-1}(v)=(u^0,u^1,u^2,u^3)$, set $F:=f\circ x^{-1}$, the smooth coordinate representative of $[f]$ near $x(p)$ [L2]. Then $f(c(t))=F(x(p)+tu)$ for $t$ near $0$, so by the chain rule [L4] applied to $t\mapsto x(p)+tu$ and to $F$, whose total derivative at $x(p)$ has matrix the Jacobian by [L3], ${\mathrm{\Phi }}_p(v)([f])=\frac{\mathrm{d}}{\mathrm{d}t}{\mid }_{0}F(x(p)+tu)=DF(x(p))\cdot u={\sum }_{a=0}^3{u}^a{\mathrm{\partial }}_{a}F(x(p))$.

${\mathrm{\Phi }}_p$ is linear in $v$: by step 2.1 the value ${\mathrm{\Phi }}_p(v)([f])={\sum }_a{u}^a{\mathrm{\partial }}_{a}F(x(p))$ is a linear function of $u=E^{-1}(v)$, and $E^{-1}$ is linear by [L6].

${\mathrm{\Phi }}_p(v)$ is a derivation at $p$: it is $\mathbb{R}$-linear in the germ because the sum and scalar-multiple derivative rules of [L7] apply to $t\mapsto f(p+tv)$ near $0$. For smooth germs $[f],[g]$, the coordinate representatives satisfy $((fg)\circ x^{-1})=(f\circ x^{-1})(g\circ x^{-1})$, so the one-variable product rule [L7] applied to the differentiable functions $t\mapsto F(y+t{\mathbf{e}}_a)$ and $t\mapsto G(y+t{\mathbf{e}}_a)$, where $y=x(p)\in {\mathbb{R}}^4$ and ${\mathbf{e}}_a\in {\mathbb{R}}^4$ is the $a$-th standard coordinate vector gives the product rule for the partial derivatives: ${\mathrm{\partial }}_a((fg)\circ x^{-1})(x(p))=f(p){\mathrm{\partial }}_a(g\circ x^{-1})(x(p))+g(p){\mathrm{\partial }}_a(f\circ x^{-1})(x(p))$, since $f(x^{-1}(x(p)))=f(p)$ and likewise for $g$; substituting into step 2.1 gives ${\mathrm{\Phi }}_p(v)([f][g])=f(p){\mathrm{\Phi }}_p(v)([g])+g(p){\mathrm{\Phi }}_p(v)([f])$, so ${\mathrm{\Phi }}_p(v)\in T_{p}A$ by [L1].

${\mathrm{\Phi }}_p$ is injective: suppose ${\mathrm{\Phi }}_p(v)=0$ with coordinate vector $u=E^{-1}(v)$. Applying ${\mathrm{\Phi }}_p(v)$ to the germ of the coordinate function $x^a$ gives, by step 2.1, ${\mathrm{\Phi }}_p(v)([x^a])={\sum }_b{u}^b{\mathrm{\partial }}_b(x^a\circ x^{-1})(x(p))={\sum }_b{u}^b{\delta }_b^a=u^a$; hence $u=0$, and since $E^{-1}$ is injective by [L6], $v=0$.

${\mathrm{\Phi }}_p$ carries the basis $(e_0,e_1,e_2,e_3)$ of $V$ to the coordinate derivations: for $v=e_a$ one has $u=E^{-1}(e_a)=({\delta }_b^a{)}_b$ by [L6], so step 2.1 gives ${\mathrm{\Phi }}_p(e_a)([f])={\mathrm{\partial }}_{a}F(x(p))={\mathrm{\partial }}_a{\mathrm{\mid }}_p([f])$ for every smooth germ, that is, ${\mathrm{\Phi }}_p(e_a)={\mathrm{\partial }}_a{\mathrm{\mid }}_p$. By [L5] the derivations ${\mathrm{\partial }}_a{\mathrm{\mid }}_p$ form a basis of $T_{p}A$, so the linear map ${\mathrm{\Phi }}_p$ carries a basis of $V$ to a basis of $T_{p}A$; hence ${\mathrm{\Phi }}_p$ is surjective (its image contains a basis) and injective by step 3.3, so it is a linear isomorphism $V\to T_{p}A$.

Finally, let $v={\sum }_a{v}^a{e}_a$; then $u=E^{-1}(v)=(v^0,v^1,v^2,v^3)$, so step 2.1 gives ${\mathrm{\Phi }}_p(v)([f])={\sum }_a{v}^a{\mathrm{\partial }}_{a}F(x(p))$, that is, ${\mathrm{\Phi }}_p(v)={\sum }_a{v}^a{\mathrm{\partial }}_a{\mathrm{\mid }}_p$. The construction uses only the affine structure $p+tv$ and the smooth structure of the model, both fixed independently of any choice of chart, so the isomorphism ${\mathrm{\Phi }}_p$ is canonical.