The important point of the proof either of these objects can be locally trivialized with transition functions on each double overlap given by a constant element of GL(n). So, given a local system, you just build the vector bundle with flat connection that has the same transition functions, and vice versa.

EDIT: Brian Conrad points out below that while this is a fairly complete sketch in the smooth case, it requires more work in the singular case.

The important point of the proof is that either of these objects can be locally trivialized with transition functions on each double overlap given by a constant element of GL(n). So, given a local system, you just build the vector bundle with flat connection that has the same transition functions, and vice versa.

EDIT: Brian Conrad points out below that while this is a fairly complete sketch in the smooth case, it requires more work in the singular case.

The important point of the proof either of these objects can be locally trivialized with transition functions on each double overlap given by a constant element of GL(n). So, given a local system, you just build the vector bundle with flat connection that has the same transition functions, and vice versa.

The important point of the proof either of these objects can be locally trivialized with transition functions on each double overlap given by a constant element of GL(n). So, given a local system, you just build the vector bundle with flat connection that has the same transition functions, and vice versa.

EDIT: Brian Conrad points out below that while this is a fairly complete sketch in the smooth case, it requires more work in the singular case.