Matrix functions have many applications in science, engineering, and the social sciences. Computation of matrix function \(f(A)\) explicitly is subject to perturbations. Therefore, it is important to understand the sensitivity of matrix functions to perturbations, which can be measured by condition numbers. The existing theory of condition number does not take into account the case, where \(A\) is structured. In this work, we focus on computing the structured condition number of matrix functions defined between smooth matrix manifolds. Since we restrict the perturbations to a smaller set we show that the structured and unstructured condition numbers can differ by several orders of magnitude, that motivates the development of algorithms preserving structure for matrix functions.