I get stuck with Remark 25.1.3.7 on Lurie's Spectral Algebraic Geometry. For sake of completeness, let's fix some notations.
Given a (discrete) commutative ring R and denote by $\operatorname{SCA}_R:=\operatorname{Fun}^{\pi}(\operatorname{Poly}_R^{\operatorname{op}},\mathcal S)$ the non-abelian derived category of the category $\operatorname{Poly}_R$ of $R$-polynomials.
This is the $\infty$-category of simplicial commutative algebras over $R$.
Now given a $R$-algebra morphism $f\colon R[x_1,\dots,x_n]\to R[y_1,\dots,y_m]$ given by $x_i\mapsto f_i(y_1,\dots,y_m)$ and an object $A\in\operatorname{SCA}_R$
We have a map of spaces induced by $f$: $A(R[y_1,\dots,y_m])\to A(R[x_1,\dots,x_n])$
In the previous remark, he identifies $A(R[x_1,\dots,x_n])$ with $A(R[X])^n$.
Now he claims that under this identification, taking the homotopy groups of the previously constructed map of spaces, we get a map $\pi_*(A)^m\to\pi_*(A)^n$, and for $*=0$, the map is given by $(a_1,\dots,a_m)\mapsto(f_1(a_1,\dots,a_m),\dots,f_n(a_1,\dots,a_m))$.
and for $*>0$ (and we assume that these spaces are pointed at $(a_1,\dots,a_m)$ and $(f_1(a),\dots,f_n(a))$), the map is given by the Jacobian matrix of $f$ at $(a_1,\dots,a_m)$.