Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$. We assume that there exist a holomorphic line bundle $K$ on $X$ and an isomorphism $$K^{\otimes 2} \cong L.$$ Show that the map $$\varphi:K \to L, \ \ (x,t) \mapsto (x,t^2)$$ is a holomorphic map.
It's been a while since I've done anything related to bundle maps, but I have a feeling that this can be proven locally using local trivializations and transition functions, but I'm not able to devise a plan on how to use these.
Could someone provide me guidance on how to use these tools to prove the statement?