  The solution of Kardar-Parisi-Zhang equation (KPZ equation) is solved
formally via Cole-Hopf transformation $h=\log u$, where $u$ is the solution of
multiplicative stochastic heat equation(SHE). In earlier works by Chatterjee
and Dunlap, Caravenna, Sun, and Zygouras, and Gu, they consider the solution of
two dimensional KPZ equation via the solution $u_\varepsilon$ of SHE with flat
initial condition and with noise which is mollified in space on scale in
$\varepsilon$ and its strength is weakened as $\beta_\varepsilon=\hat{\beta}
\sqrt{\frac{2\pi \varepsilon}{-\log \varepsilon}}$, and they prove that when
$\hat{\beta}\in (0,1)$, $\frac{1}{\beta_\varepsilon}(\log
u_\varepsilon-\mathbb{E}[\log u_\varepsilon])$ converges in distribution to a
solution of Edward-Wilkinson model as a random field.
  In this paper, we consider a stochastic heat equation $u_\varepsilon$ with
general initial condition $u_0$ and its transformation $F(u_\varepsilon)$ for
$F$ in a class of functions $\mathfrak{F}$, which contains $F(x)=x^p$ ($0<p\leq
1$) and $F(x)=\log x$. Then, we prove that
$\frac{1}{\beta_\varepsilon}(F(u_\varepsilon(t,x))-\mathbb{E}[F(u_\varepsilon(t,x))])$
converges in distribution to Gaussian random variables jointly in finitely many
$F\in \mathfrak{F}$, $t$, and $u_0$. In particular, we obtain the fluctuations
of solutions of stochastic heat equations and KPZ equations jointly converge to
solutions of SPDEs which depends on $u_0$.
  Our main tools are It\^o's formula, the martingale central limit theorem, and
the homogenization argument as in the works by Cosco and the authors. To this
end, we also prove the local limit theorem for the partition function of
intermediate $2d$-directed polymers
