To be honest, I don't know what to think about Carrot. I know that some here have viewed him as Terry's early watch-hero in G!G! and MAA, but he never struck me that way. I always viewed Vimes as the hero, and Carrot as second fiddle at best.
I know that in MAA, he and Angua form the basis of their later "friendship" (ahem
courtship), but this never made him the hero for me, even in MAA. Yes, Vimes was going to retire because he was getting married, but so what?

I knew that Vimes wouldn't stay retired for long, because:
a) He was the anti-hero of G!G! (like Batman is the anti-hero of his eponymous comic books/films/whatever), and no author throws away a good anti-hero; and
b) see a).
But although Carrot was never the hero, for me, he was a perfectly good second banana.* He was truly good, trustworthy, strong, etc. etc. - all the things, in fact, that the Elucidated Brotherhood of the Ebon Night
didn't want in a king (or rather, the Grand Master didn't; there's no way to know if the Brotherhood did or didn't. Lumps of charcoal seldom have much to say on monarchy).
The problem with Carrot in G!G! was that he was too perfect - or rather, that he succeeded in spite of his weaknesses (i.e. his recklessness and ingenuousness). Yes, Vimes also succeeds in the end in spite of his weaknesses (i.e. his "dark side" and addiction to liquor), but Carrot succeeds
right away in everything he does, and that's not good for character development.
A character ought to face conflicts, which Carrot does, and he ought to face setbacks - which Carrot almost never does. It's the setbacks that teach a character the most and allow him to react to them, because they create a dilemma the character has to face, and thus make a decision.
These setbacks, dilemmas and decisions help the character to grow and develop, but Carrot almost never develops - it's not until "The Fifth Elephant" when we see how much he cares about Angua, when he resigns his commission and sets out after her. But this decision goes against Carrot's philosophy that we've in the past four books: "Personal is not the same thing as important". Now it's suddenly VERY important. So what's changed? It's never explained, which is very frustrating.
Don't get me wrong: Carrot's simplicity and honesty make him a very funny character. His bravery in the face of danger is endearing. His readiness to face overwhelming odds is inspiring. But his basic philosophy simply doesn't change - until TFE, when he suddenly
does change because Angua's gone to Uberwald. Why should he chase her? Because she's gone to Uberwald? Perhaps his personal desires
are important after all. But then, at the end of the book, he changes back when he says that, by putting his personal desires ahead of his duty to the watch and to A-M, he's failed.
And the ending is puzzling: after he's chased away Colon and Nobby to get the rest of the watchmen again, he and Angua seem to have a "moment". "Wolves never look back," he whispers. Does that mean that, in spite of everything, he
has changed - and wants to be as devoted to her as he is to the City?
Yet by the time we see them next, in NW and TLH, their relationship is almost not mentioned. I think that, in MR, it's mentioned that they're in a stable relationship; how it got that way, though, is never explained. I think we all need to use our imaginations.
So, yes. Carrot: the problematic, frustrating man-child of the Watch who never grows up? Or the endearing, funny, inspiring bloke, who keeps his duty strictly before his eyes, in the best traditions of the Victorian age?
You decide.
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* kids, if you don't know what that is, ask your dad. He'll fill you in.