Let $p_1:V\rightarrow X$ and $p_2:W\rightarrow X$ be two algebraic vector bundle over a variety $X$ of rank $n$ and $m$ respectively. Let's say $\{U_i,f_i\}$ and $\{U_i,g_i\}$ are trivializations for the vector bundles $V$ and $W$ respectively.
We define, $Mor(V,W)=\amalg_{x\in X}Hom(V_x,W_x)$.
Let $p:Mor(V,W)\rightarrow X$ be defined by $Hom(V_x,W_x)\rightarrow x$
The next step is to define local trivialization (i.e. we need to define a map from $\phi_i:p^{-1}(U_i)\rightarrow U_i\times GL(nm, \mathbb{C})$ such that $proj\circ\phi_i=p$, where $proj: U_i\times GL(nm, \mathbb{C})\rightarrow U_i$ is the projection map)
Notice that, $p^{-1}(U_i)=\amalg_{x\in U_i}Hom(V_x,W_x)$
The map $\phi_i:\amalg_{x\in U_i}Hom(V_x,W_x)\rightarrow U_i\times GL(nm, \mathbb{C})$ can be defined as $Hom(V_x,W_x)\rightarrow x\times \text{The matrix correspinding to the map from}$ $V_x$ to $W_x$.
But I have a feeling that the definition of should involve $f_i$ and $g_i$ (the local trivializaions for $V$ and $W$). Also, there is no guarantee that the matrix corresponding to the map form $V_x$ to $W_x$ will be non-singular, and the map I defined will not be well-defined.
My main aim is to write down the transition functions for the vector bundle $Mor(V,W)$ But I am stuck far before that in giving a vector bundle structure to $Mor(V,W)=\amalg_{x\in X}Hom(V_x,W_x)$.
If $V$ is defined using transition functions $f_{ij} : U_i \cap U_j \to GL_n(\mathbb{C})$, thought of as acting on $\mathbb{C}^n$, and $W$ is defined using transition functions $g_{ij} : U_i \cap U_j \to GL_m(\mathbb{C})$, thought of as acting on $\mathbb{C}^m$ then the hom bundle $[V, W]$ is defined using transition functions
$$T \mapsto g_{ij} T f_{ij}^{-1}$$
thought of as acting on $[\mathbb{C}^n, \mathbb{C}^m]$.
Similarly, the tensor product bundle $V \otimes W$ is defined using transition functions $f_{ij} \otimes g_{ij} : U_i \cap U_j \to GL_{nm}(\mathbb{C})$, thought of as acting on $\mathbb{C}^n \otimes \mathbb{C}^m$.