Let $f:X\rightarrow Y$ be a bijection map between complex varieties, when will it be a morphism?

I meet this question when working over Fourier-Mukai transformas in algebraic geometry and some papers. In Corollary 5.23 of the book above, it induces a morphism from a bijection $f$ between smooth projective complex varieties as following.

Let $P$ be a $X$-flat sheaf on $X\times Y$ such that $P|_{\{x\}\times Y}\cong k(f(x))$ is the skyscraper sheaf for each closed point $x\in X$, then $f$ induces a morphism $X\rightarrow Y$.

How does it work? Also, I meet in some papers in complex algebraic geometry that people define a morphism by claiming on closed points. Is it something about GAGA?

Let $f:X\rightarrow Y$ be a bijective map between complex varieties, when will it be a morphism?

I meet this question when working over Fourier–Mukai transforms in algebraic geometry and some papers. In Corollary 5.23 of the book above, it induces a morphism from a bijection $f$ between smooth projective complex varieties as follows.

Let $P$ be a $X$-flat sheaf on $X\times Y$ such that $P\rvert_{\{x\}\times Y}\cong k(f(x))$ is the skyscraper sheaf for each closed point $x\in X$. Then $f$ induces a morphism $X\rightarrow Y$.

How does it work? Also, I meet in some papers in complex algebraic geometry that people define a morphism by claiming on closed points. Is it something about GAGA?

Let $f:X\rightarrow Y$ be a bijection map between complex varieties, when will it be a morphism?

I meet this question when working over Fourier-Mukai transformas in algebraic geometry and some papers. In Corollary 5.23 of the book above, it induces a morphism from a bijection $f$ between smooth projective complex varieties as following.

Let $P$ be a $X$-flat sheaf on $X\times Y$ such that $P|_{\{x\}\times Y}\cong k(f(x))$ is the skyscraper sheaf for each closed point $x\in X$, then $f$ induces a morphism $X\rightarrow Y$.

How does it work? Also, I meet in some papers in complex geometry that people define a morphism by claiming on closed points. Is it always reasonable? or we need to clarify.

Let $f:X\rightarrow Y$ be a bijection map between complex varieties, when will it be a morphism?

I meet this question when working over Fourier-Mukai transformas in algebraic geometry and some papers. In Corollary 5.23 of the book above, it induces a morphism from a bijection $f$ between smooth projective complex varieties as following.

Let $P$ be a $X$-flat sheaf on $X\times Y$ such that $P|_{\{x\}\times Y}\cong k(f(x))$ is the skyscraper sheaf for each closed point $x\in X$, then $f$ induces a morphism $X\rightarrow Y$.

How does it work? Also, I meet in some papers in complex algebraic geometry that people define a morphism by claiming on closed points. Is it something about GAGA?

Let $f:X\rightarrow Y$ be a bijection map between complex varieties, when will it be a morphism?

I meet this question when working over Fourier-Mukai transformas in algebraic geometry and some papers. In Corollary 5.23 of the book above, it induces a morphism from a bijection $f$ between smooth projective complex varieties as following.

Let $P$ be a $X$-flat sheaf on $X\times Y$ such that $P|_{\{x\}\times Y}\cong k(f(x))$ is the skyscraper sheaf for each closed point $x\in X$, then $f$ induces a morphism $X\rightarrow Y$.

How does it work? Also, I meet in some papers in complex geometry that people define a morphism by claiming on closed points. Is it always reasonable or we need to clarify.

Let $f:X\rightarrow Y$ be a bijection map between complex varieties, when will it be a morphism?

I meet this question when working over Fourier-Mukai transformas in algebraic geometry and some papers. In Corollary 5.23 of the book above, it induces a morphism from a bijection $f$ between smooth projective complex varieties as following.

Let $P$ be a $X$-flat sheaf on $X\times Y$ such that $P|_{\{x\}\times Y}\cong k(f(x))$ is the skyscraper sheaf for each closed point $x\in X$, then $f$ induces a morphism $X\rightarrow Y$.

How does it work? Also, I meet in some papers in complex geometry that people define a morphism by claiming on closed points. Is it always reasonable? or we need to clarify.