Holomorphic tangent bundle can be defined by cocycle of holomorphic Jacobians of transition maps. But this method will give different bundles which only agree up to isomorphism. I see in some text that show the holomorphic tangent bundle of a complex submanifold is naturally a subbundle of the holomorphic tangent bundle of the ambient complex manifold. But the argument confuse me like other categorical constructions. The author "choose" a local holomorphic chart such that the submanifold embedded flatly. Why he doesnt check the effect of choose? I know this case can be save by two methods:
1 Check another choose give an essential same map;
2 Show the holomorphic tangent bundle has an invariant description by almost complex structure.
I cant understand the attitude of completely ignoring the checking. Furthermore if you can choose arbitrary cocycle, then you get different bundles and different maps, and in fact you define nothing. I wonder if we should fix a choice in a definition.
More generally, any categorical constructions will face similar case. E.g, when you define the kernels of maps between universal objects and consider their products, even compute some more complex constructions, why people never consider the agreement problem via different choose?