  In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a semigroup
$e^{-tA_\varepsilon}$, $t\geqslant 0$, generated by a matrix elliptic second
order differential operator $A_\varepsilon \geqslant 0$. Coefficients of
$A_\varepsilon$ are periodic, depend on $\mathbf{x}/\varepsilon$ and oscillate
rapidly as $\varepsilon \rightarrow 0$. Approximations for
$e^{-tA_\varepsilon}$ were obtained by T. A. Suslina (2004, 2010) via the
spectral method and by V. V. Zhikov and S. E. Pastukhova (2006) via the shift
method. In the present note, we give another short proof based on the contour
integral representation for the semigroup and approximations for the resolvent
with two-parametric error estimates obtained by T. A. Suslina (2015).
