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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.

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