The remarkable ability of transformers to learn new concepts solely by reading examples within the input prompt, termed in-context learning (ICL), is a crucial aspect of intelligent behavior. Here, we focus on understanding the learning algorithm transformers use to learn from context. Existing theoretical work, often based on simplifying assumptions, has primarily focused on linear self-attention and continuous regression tasks, finding transformers can learn in-context by gradient descent. Given that transformers are typically trained on discrete and complex tasks, we bridge the gap from this existing work to the setting of classification, with non-linear (importantly, softmax) activation. We find that transformers still learn to do gradient descent in-context, though on functionals in the kernel feature space and with a context-adaptive learning rate in the case of softmax transformer.