The categories of sequential spaces and compactly generated spaces both use a finer product than the one from $\textbf{Top}$ in order to be cartesian closed, and in both cases the product is arguably more natural as well, especially in the sequential case.
Every sequential space is (Hausdorff-) compactly generated. This can be seen either indirectly from it being a quotient of a metric space, or more explicitly by viewing sequential spaces as generated by maps from the compact (Hausdorff) space $\mathbb{N}\cup{\{\infty\}}$ that characterize convergent sequences.
Does the product of sequential spaces coincide with the one from compactly generated spaces? I'm interested both in the case of finite products and infinite products / exponentials.
Since every sequential space is compactly generated, abstract nonsense quickly shows that the sequential product topology is finer than or equal to the compactly generated product topology, but the other direction is harder. So it can be reduced to whether the (compact closed) product of sequential spaces is sequential.