Question about equivalent definitions of (holomorphic) line bundles

Definition: A complex line bundle is a smooth manifold $$L$$ with a projection map $$\pi:L\to M$$, so that each fiber $$\pi^{-1}(p)$$ has the structure of a complex vector space of dimension 1. And for each $$p\in M$$, there is a neighborhood $$U$$ on which $$L$$ trivializes, i.e. there exists a homeomorphism $$\varphi_U:\pi^{-1}(U)\to M$$ so that $$\varphi_U(\eta)=(\pi(\eta),\hat\varphi_U(\eta))$$, where $$\varphi_{\pi^{-1}(\eta)}$$ is a $$\mathbb C$$-linear isomorphism.

We may define the transition maps $$g_{UV}:U\cap V\to\mathbb C$$ for local trivializations $$U,V$$. It is said that the family of transition maps also uniquely determine a line bundle (under certain conditions on the transition functions, e.g. $$g_{UV}=g_{VU}^{-1}$$), by identifying corresponding points in $$U\times\mathbb C$$ and $$V\times\mathbb C$$. This gives an alternative definition of complex line bundle.

Question $$1$$: How do know when two sets of transition maps determine the same complex line bundle?

We may go on defining a holomorphic line bundle to be a complex line bundle so that the transition maps for an open cover $$\{U_j\}_J$$ are holomorphic maps.

Question $$2$$: How do we show that if $$U$$ is some other open set on which $$L$$ trivializes, then the transition map $$g_{U_jU}$$ are also holomorphic?

For completeness, I answer my own question. Since I am still a beginner, I will write in full details. There is no original insights in the following. In fact the original questions/confusions came from not fully understanding the notions/constructions. And such confusions are resolved once I make clear of the statements/definitions. Any criticisms/clarifications are welcomed.

Let $$M$$ be a complex manifold, let $$L$$ be a line bundle on $$M$$, equipped with the projection $$\pi:L\to M$$.

The transition functions arise in the following way: Let $$\{U_i\} _I$$ be an open cover of $$M$$ so that $$U_i$$ are local trivialization of $$L$$, meaning that there is a diffeomorphism $$\varphi_i:\pi^{-1}(U_i)\to U_i\times\mathbb C$$ so that $$\varphi_i(\eta) =(\pi(\eta), \hat\varphi_i(\eta))$$, where $$\hat\varphi_i$$ are linear isomophism on each fiber. Then $$\varphi_j\circ\varphi_k^{-1}(x,\lambda)=(x, \hat\varphi_i\circ\hat\varphi_j^{-1}(\lambda))$$, whenever defined. And we have for each fixed $$x\in U_j\cap U_k$$, $$\lambda \mapsto \hat\varphi_j\circ\hat\varphi_k^{-1}(\lambda)$$ is a linear isomorphism and is given by a constant depending on $$x$$. This gives rise to the transition functions satisfying $$(x, g_{U_jU_k}(x)\lambda)= \varphi_j\circ\varphi_k^{-1}(x,\lambda)$$.

Conversely, suppose $$\{U_i\}_I$$ is an open cover, given a family of transition functions $$g_{UV}$$ for each pair of open sets $$U, V$$, satisfying $$g_{UV}g_{VW} =g_{UW}$$ whenever defined, note that this implies that $$g_{UU}\equiv 1$$ for any $$U$$ and $$g_{UV} =\frac1{g_{VU} }$$. We may define the line bundle $$L=\bigsqcup_I U_i\times\mathbb C/\sim$$, where $$(x_j, \lambda_j) \sim (x_k, \lambda_k)$$ iff $$x_j=x_k$$ and $$g_{U_jU_k}(x_k)\lambda_k=\lambda_j$$. Denote the equivalent class of $$(x, \lambda)$$ by $$[(x, \lambda)]$$, the projection map is given by $$\pi([(x, \lambda)]) =x$$. The claim is that it is always possible to construct local trivialization on each $$U_i$$ so that the transition functions is precisely given by the $$g$$'s. To see this, for any $$U_i$$, we have $$\pi^{-1}(U_i)=[U_i\times\mathbb C]$$. Since for any $$[(x, \lambda)] \in [U_i\times\mathbb C]$$, there is precisely one representative from $$U_i\times\mathbb C$$, so we may define the map $$\varphi_{U_i}:\pi^{-1}(U_i)\to U_i\times\mathbb C$$ by $$\varphi_{U_i}([(x, \lambda)] = (x_i, \lambda_i)$$ for the unique representative, the claim follows.

Now answering the questions: for two open covers $$\mathcal U_1,\mathcal U_2$$ and two families of transition functions, we can construct corresponding line bundles $$L_1,L_2$$ as outlined above. If for each pair of open sets $$U$$ and $$V$$ in their respective open covers, we can a transition function $$g_{UV}$$ that is compatible with the original families of transition functions, then the two bundles must be isomorphic. Since we may use the open cover $$\mathcal U_1\cup\mathcal U_2$$ and put together the family of transition functions to form a third line bundle $$L_3$$, this line bundle would be isomorphic to both $$L_1$$ and $$L_2$$, as we may define the fiber preserving morphism $$f:L_1\to L_3$$ by $$f([(x, \lambda)]_1) =[(x, \lambda)]_3$$, which admits an inverse by mapping $$\eta=[(x, \lambda)]_1 \in L_3$$ to the unique equivalent class of $$L_1$$ that is contained in $$\eta$$ as subset. Now we may refine the open covers in the following way, define $$\mathcal U=\{U_1\cap U_2|U_1\in\mathcal U_1, U_2\in\mathcal U_2\}$$. We may define transition functions on the refined open cover according to the transition functions we started with. We see by the same argument that the line bundles obtained from the refinements are actually isomorphic to the original bundles $$L_i$$, provided that the transition functions are homotopic, as described in this question.

The second question may be answered similarly. Again we construct the line bundle $$L$$ as a quotient. The idea is that for holomorphic transition functions, we can define a complex structure on $$L$$ via the charts $$(h_i\times\operatorname{Id})\circ\varphi_{U_i}$$ where $$h_j$$ is a chart on $$U_i\subset M$$ (in full details we need to find another open cover consisting of charts and intersect with the original cover, but we may assume the original cover already consists of charts using refinement again). If $$U$$ is other open set over which $$L$$ trivializes, then by definition we have a biholomorphism $$\varphi_U: \pi^{-1}(U)\to U\times\mathbb C$$. If $$U_i, U_j$$ is any open set in the open cover that intersects with $$U$$, then $$(h_j\times\operatorname{Id})\circ \varphi_U\circ\varphi_{U_i}^{-1}\circ(h_i^{-1}\times\operatorname{Id})(x, \lambda) =(h_j(h_i^{-1}(x)), g_{UU_i}( h_j^{-1} (x)) \lambda)$$ is a biholomorphism. Hence $$g_{UU_i}$$ must be a holomorphic function.

This approach may be generalized to vector bundles.