  We study numerically the two-dimensional Ising model with non-conserved
dynamics quenched from an initial equilibrium state at the temperature $T_i\ge
T_c$ to a final temperature $T_f$ below the critical one. By considering
processes initiating both from a disordered state at infinite temperature
$T_i=\infty$ and from the critical configurations at $T_i=T_c$ and spanning the
range of final temperatures $T_f\in [0,T_c[$ we elucidate the role played by
$T_i$ and $T_f$ on the aging properties and, in particular, on the behavior of
the autocorrelation $C$ and of the integrated response function $\chi$. Our
results show that for any choice of $T_f$, while the autocorrelation function
exponent $\lambda _C$ takes a markedly different value for $T_i=\infty$
[$\lambda _C(T_i=\infty)\simeq 5/4$] or $T_i=T_c$ [$\lambda _C(T_i=T_c)\simeq
1/8$] the response function exponents are unchanged. Supported by the outcome
of the analytical solution of the solvable spherical model we interpret this
fact as due to the different contributions provided to autocorrelation and
response by the large-scale properties of the system. As changing $T_f$ is
considered, although this is expected to play no role in the
large-scale/long-time properties of the system, we show important effects on
the quantitative behavior of $\chi$. In particular, data for quenches to
$T_f=0$ are consistent with a value of the response function exponent $\lambda
_\chi=\frac{1}{2}\lambda _C(T_i=\infty)=5/8$ different from the one [$\lambda
_\chi \in (0.5-0.56)$] found in a wealth of previous numerical determinations
in quenches to finite final temperatures. This is interpreted as due to
important pre-asymptotic corrections associated to $T_f>0$.
