Vtyjkyuk

I'm studying the relative tangent spaces in Algebraic Geometry I by Gortz,Wedhorn. The definition as follows:

Let $X$ be a $S$-scheme, $K$ be a filed and $xi : mathrmSpecK rightarrow X$ be a $K$-valued point of $X$. We define the relative tangent space $T_xi(X/S)$ of $X$ in $xi$ over $S$ as the set of $S$-morphisms $t:mathrmSpecK[varepsilon] rightarrow X$ such that the composition of t with $mathrmSpecK rightarrow mathrmSpecK[varepsilon]$ is equal to $xi$ , where $K[varepsilon]$ is the ring of dual number over K.

Remark 6.12. (2) of this book says:

Let $iota : K hookrightarrow L$ be a field extnsion corresonding to the morphism $p:mathrmSpecL rightarrow mathrmSpecK$ and let $q:mathrmSpecL[varepsilon] rightarrow mathrmSpecK[varepsilon]$ be the canonical morphism induced by $p$. Then a morphism $phi : T_xi(X/S) otimes_KL rightarrow T_xi circ p(X/S)$ ,$totimes ell mapsto ell cdot(t circ q) $ is an isomorhism of $L$-vector spaces.

I can show that $phi$ is injective. But Is this surjective? Or Can I construct
the inverse map of $phi$ ?

Remark: $X$ is a "general" $S$-scheme. So the relative tangent space $T_xi(X/S)$ of X is not necessarily finite dimensional.