I find it easier to understand in terms of the Unix syscall API. `2>&1` literally translates as `dup2(1, 2)`, and indeed that's exactly how it works. In the classic unix shells that's all that happens; in more modern shells there may be some additional internal bookkeeping to remember state. Understanding it as dup2 means it's easier to understand how successive redirections work, though you also have to know that redirection operators are executed left-to-right, and traditionally each operator was executed immediately as it was parsed, left-to-right. The pipe operator works similarly, though it's a combination of fork and dup'ing, with the command being forked off from the shell as a child before processing the remainder of the line.
Though, understanding it this way makes the direction of the angled bracket a little odd; at least for me it's more natural to understand dup2(2, 1) as 2<1, as in make fd 2 a duplicate of fd 1, but in terms of abstract I/O semantics that would be misleading.
Though, understanding it this way makes the direction of the angled bracket a little odd; at least for me it's more natural to understand dup2(2, 1) as 2<1, as in make fd 2 a duplicate of fd 1, but in terms of abstract I/O semantics that would be misleading.