Base extension of the relative tangent spaces

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 $\xi : \mathrm{Spec}K \rightarrow X$ 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 $t:\mathrm{Spec}K[\varepsilon] \rightarrow X$ such that the composition of t with $\mathrm{Spec}K \rightarrow \mathrm{Spec}K[\varepsilon]$ 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 $p:\mathrm{Spec}L \rightarrow \mathrm{Spec}K$ and let $q:\mathrm{Spec}L[\varepsilon] \rightarrow \mathrm{Spec}K[\varepsilon]$ be the canonical morphism induced by $p$. Then a morphism $\phi : T_{\xi}(X/S) \otimes_{K}L \rightarrow T_{\xi \circ p}(X/S)$ ,$t\otimes \ell \mapsto \ell \cdot(t \circ q)$ 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.