  We establish large deviation principle (LDP) for the family of vector-valued
random processes $(X^\epsilon,Y^\epsilon),\epsilon\to 0$ defined as $$
X^\epsilon_t=\frac{1}{\epsilon^\kappa}\int_0^t
H(\xi^\epsilon_s,Y^\epsilon_s)ds,
dY^\epsilon_t=F(\xi^\epsilon_t,Y^\epsilon_t)dt+
D\epsilon^{1/2-\kappa}G(\xi^\epsilon_t,Y^\epsilon_t)dW_t,$$ where $W_t$ is
Wiener process and $\xi^\epsilon_t$ is fast ergodic diffusion. We show that,
under $\kappa<{1/2}$ or less and Veretennikov-Khasminskii type condition for
fast diffusion, the LDP holds with rate function of Freidlin-Wentzell's type.
