In the move sequence I show here:
https://online-go.com/demo/1412286/13

white a mistake on move 5... however, I'm wondering about move 13 by Black. Why did he form this kosumi? Was it defensive or offensive?

Why not A, B, or C as shown in the diagram?

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What can Black do afterwards? There is O13 (d). So White will probably have to somehow defend here now. After a few exchanges Black can play F17 or something, but has probably gained a few more stones facing the upper side and/or the lower centre.

If Black played A (a) instead, White is under far less pressure and maybe has time to play C (c), or at least a somewhat more active move in the centre, e. g. L13 (jump to e).

If Black played B (b) or C (c), White could defend the group more actively, e. g. M15 (f), which limits Black's development at the top. Black would probably even have to answer, because otherwise a White followup at M17 (g) would make the black corner very thin.

Click Here To Show Diagram Code

[go]$$
$$ ----------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . b . . . . . g . . . . . . . |
$$ | . . . O . . . . . a . . . X . X . . . |
$$ | . . . . . . . . . . . f B . . . X . . |
$$ | . . . . . . . . . . . . . . . O O . . |
$$ | . . . . . . . . . . e . O d . . . . . |
$$ | . . . . . . . . . . . . . . . O . . . |
$$ | . . . . . . . . . . . . . X . . . . . |
$$ | . . . , . . . . . , . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . c . . . . . X . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ----------------------------------------[/go]

(post edited to be self-contained)

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A good system naturally covers all corner cases without further effort.

It seems unfair toward the community to post the same question here and on reddit at the same time. At least I think so.

Also, there is no diagram, only a link.

Is the overall position in the link from Michael Redmond's book? GoGoD has no example from professional play of White playing such an early inside approach. In the Shin Fuseki period there were many people who played on the star point on the left side (about a hundred games). Later people almost invariably approached from the outside (over a thousand games).

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Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

I'm still learning, but it looks to me like move 13 by Black was defensive. If Black played A, B, or C, White could potentially cut through and create a weak group. The kosumi strengthens Black's position.