At first, Saigo didn't really know what to make of Kyojuro. His bright eyes and boldness not only confused him, but disturbed him to some degree.
Kyojuro was the first one to show empathy for Saigo's trauma. Because of his name, many of the demon slayers would murmur about how he was a posh, wealthy child who had run away from home. But Kyojuro gave him the benefit of the doubt and Saigo was grateful for that.