Let \(\varphi\) and \(\lambda\) be the Euler and Carmichael functions, respectively. In this paper, the authors establish lower and upper bounds for the counting function of the set \(\mathcal A(x) = \{n\leq x\mid \varphi(\lambda(n))=\lambda(\varphi(n))\}\). The main results are the following: Theorem 1. There exist positive constants \(C\) and \(x_0\) such that the following bound holds for all \(x\geq x_0\): \[ \#\mathcal A(x)\geq \exp \left(C \frac{\log x}{\log\log x}\right). \] Theorem 2. The inequality \[ \#\mathcal A(x)\leq \frac{x}{(\log x)^{3/2+o(1)}} \] holds as \(x\to\infty\). They also study the normal order of the function \(\varphi(\lambda(n))/\lambda(\varphi(n))\). Theorem 3: The estimate \[ \frac{\varphi(\lambda(n))}{\lambda(\varphi(n))}=\exp\bigl((1+o(1))(\log\log n)^2\log\log\log n\bigr) \] holds on a set of positive integers \(n\) of asymptotic density one. In particular, one sees that \(\varphi(\lambda(n))\) is much larger than \(\lambda(\varphi(n))\) for almost all positive integers \(n\).