This is from Le Potier's Lectures on Vector Bundle
Definition: A complex linear fibration (or just fibration) over an algebraic variety is a pair $(E,p)$ where E is an algebraic variety and $p:E\rightarrow X$ is a surjective morphism such that for each point $x\in X$ there exists complex vector space structure of on the fiber $p^{-1}(x)$.
Given two fibrations $p:E\rightarrow X$ and $p':E'\rightarrow X$, by a map of linear fibrations we mean a map of algebraic varieties such $f:E\rightarrow E'$ such that it is compatible with the projection, i.e., $p'\circ f=p$.
The bundle $X\times \mathbb{C}^r\rightarrow X$ given by the projection to the first factor is called the trivial fibration of rank $r$.
For each open set $U\subset X$ and fibration $p:E\rightarrow X$ we write $E_{|}U$ for the fibration $p_{|U}:p^{-1}(U)\rightarrow U$
Algebraic Vector Bundle: An algebraic vector bundle of rank $r$ on $X$ is a linear fibration $E\rightarrow X$ which is locally trivial in the following sense: for each point $x\in X$ there exists an open neighbourhood $U$ of $x$ and an isomorphism of fibratons $\phi_U:E_{|U}\rightarrow U\times \mathbb{C}^r$er (i.e., same as saying there exist an open covering $\{U_i\}$ of $X$ and $\phi_{U_i}:E_{|U_i}\rightarrow U_i\times \mathbb{C}^r$;such an isomorphism is called a local chart or trivialization of the vector bundle on $U_i$)
By the property of morphisms of fibrations we have $p_1\circ\phi_{U_i}=p$---(*) where $p_1:U_i\times \mathbb{C}^r\rightarrow U_i$ is the projection to the first factor.
Transition functions: Let us say there are two overlapping elements in the open covering $U_i$ and $U_j$. We have two trivializations over the open set $U_i\cap U_j$:
$\phi_{U_i}$$_{|U_i\cap U_j}:E_{|U_i\cap U_j}\rightarrow (U_i\cap U_j) \times \mathbb{C}^r$ and $\phi_{U_j}$$_{|U_i\cap U_j}:E_{|U_i\cap U_j}\rightarrow (U_i\cap U_j) \times \mathbb{C}^r$
So, we get a map $ \phi_{ij}:=(\phi_{U_i}$$_{|U_i\cap U_j})\circ(\phi_{U_j}$$_{|U_i\cap U_j})^{-1}: (U_i\cap U_j) \times \mathbb{C}^r \rightarrow (U_i\cap U_j) \times \mathbb{C}^r$
By the property $(*)$ we have $p_1\circ \phi_{ij}(x,v)=x$ i.e., it induces a map $g_{ij}:U_i\cap U_j\rightarrow \{\text{to isomorphisms of }\mathbb{C}^r\}$ which is an map of algebraic varieties, this maps are called transition functions.
My question is why are the transition functions are maps of algebraic varieties?