The Cowboy, who has only two brief on screen appearances and is the film's briefly used narrator, nonetheless has a powerful impact on the film as a whole.
The brilliance of the Coen brothers is the evidenced in the varied interchange of The Dude, Walter, and Donny and interactions with the various other characters. Out of the numerous characters that cross The Dude's path, the two brief scenes with The Cowboy seem to be intentionally staged to create brief and pointed separations in the overall random and rambling pace of the plot.
Those who value acceptance may find a redeeming moment in these brief interactions. The Dude and The Cowboy, who seem to be from quite different backgrounds, are both able to instantly recognize some unexplainable personal connection.

The Cowboy:
(stopping short of saying what he was going to say and breaking into a momentary wide smile)

Just one thing, Dude... Do ya have to use s'many
cuss words?

Dude: (a full brevity shortening of "what" and "the") 'th fuck are you