Consider the following result from p. 89 of Mumford's Abelian Varieties.
Proposition. Let $X$ be a complete variety, $Y$ any scheme and $L$ a line bundle on $X \times Y$. Then there exists a unique closed subscheme $Y_1 \hookrightarrow Y$ having the following properties:
(a) if $L_1$ is the restriction of $L$ to $X \times Y_1$, there is a line bundle $M_1$ on $Y$ and an isomorphism $p_2^* M_1 \cong L_1$ on $X \times Y_1$;
(b) if $f: Z \to Y$ is any morphism such that there exists a line bundle $K$ on $Z$ and an isomorphism $p_2^*(K) \cong (1_X \times f)^*(L)$ on $X \times Z$, $f$ can be factored as $Z \to Y_1 \hookrightarrow Y$.
Is it possible somebody could give me their intuitions behind Mumford's proof of this result? The proof is as follows.
