Descent datum for a module Below is a definition of a descent datum on stacks project: ![enter image description here]( It then says that if $N = B \otimes_AM$ for some $A$-module $M$, then it has a canonical descent datum given by the map $$N \otimes_AB \to B \otimes_AN'$$ $$b_0 \otimes m \otimes b_1 \mapsto b_0 \otimes b_1 \otimes m$$ Why are modules of the form $B \otimes_AM$ special? Don't you always have a descent datum $n \otimes b \mapsto b \otimes n$? for a $B$-module $N$?

I can't see that your map $\phi:n\
ewcommand{\ot}{\otimes}\ot b\mapsto b\ot n$ is a $B\ot_A B$-module homomorphism. $$\phi((b_1\ot b_2)(n\ot b))=\phi(b_1n\ot b_2b)= b_2b\ot b_1 n$$ but $$(b_1\ot b_2)\phi(n\ot b)=(b_1\ot b_2)(b\ot n)=b_1b\ot b_2n.$$ Those look different to me.