Let $C\rightarrow B \rightarrow A$ be sequence (not exact) of surjective ring homomorphisms with $K=ker(C \rightarrow B)$, $I=ker(B \rightarrow A), J=ker(C \rightarrow A)$. I can see that we have an exact sequence $J/J^2 \rightarrow I/I^2 \rightarrow 0$ and also that $I=J/K$. But why do we have an exact sequence $K/K^2 \otimes_B A \rightarrow J/J^2 \rightarrow I/I^2 \rightarrow 0$ and why is $K/K^2 \otimes_B A$ a $B$-module? In particular, $A$ is a $B$ module, but why is $K/K^2$ a $B$-module?
The motivation of this question comes from Lemma 33.5 of the chapter of Morphisms of Schemes of the Stacks Project.
Edited: What exactly is the map $(K/K^2) \otimes_B A \rightarrow J/J^2$?