Theorem: Algebraic descent criterion

Let $v:E\to A$ be surjective and let $\lambda :E\otimes Z\to Y$ be linear. There is a unique bilinear map $\ell :A\times Z\to Y$ with

if and only if

For two presented arguments, the condition is vanishing on

Proof. Necessity follows by inserting a zero argument. For sufficiency, two presentations of the same argument differ by an element of the indicated kernel; the condition makes their assigned values equal. Linearity gives bilinearity and surjectivity gives uniqueness. Apply the argument to both slots for the last assertion.