linear algebra – Proof of the Universal Property of the Tensor Produts

Suppose that $h:Xtimes Yto mathbb{F}$ is a bilinear map. Then prove that there exists a linear map $h_otimes:Xotimes Ytomathbb{F}$ such that $h(x,y)=h_otimes(x,y)$.

Logically, to every element $(x,y)in Xtimes Y$ there is an element $xotimes yin Xotimes Y$, therefore we there can be a one to one correspondance between both the maps.

But how do I prove that $h_otimes$ is a linear map ?

Let, $h:Xtimes Ytomathbb{F}$ such that $h(x,y)=f(x)g(y)$ where $f:Xtomathbb{F}$ and $g:Ytomathbb{F}$

Note : I only have understanding of the linear algebra basics.