I was reading Professor Kawamata's paper Deformation of Canonical Singularity (which also appears in the book Algebraic Varieties: Minimal Models and Finite Generation Corollary 2.12.11) the main theorem is stated as follows:

Corollary 2.12.11.Let $f: \mathcal{X} \rightarrow B$ be a flat morphism from a germ of an algebraic variety to a germ of a smooth curve. Assume that the central fiber $X_0=f^{-1}(P)$ has only canonical singularities. Then so has the total space $\mathcal{X}$ as well as any fiber $X_t$ of $f$. Moreover, if $\mu: V \rightarrow \mathcal{X}$ is a birational morphism from a normal variety with the strict transform $X$ of $X_0$, then $K_V+X \geq \mu^*\left(K_{\mathcal{X}}+X_0\right)$.

The key technical lemma that will be used is the following extension theorem

Theorem A.Let $V$ be a smooth variety with $X$ a smooth divisor on $V$, with $\mu:V\to \mathcal{X}$. Assume that $K_V+X$ is $\mu$-big for the pair $(V, X)$. Then the natural homomorphism $\mu_* \mathcal{O}_V\left(m\left(K_V+X\right)\right) \rightarrow \mu_* \mathcal{O}_X\left(m K_X\right)$ is surjective for any positive integer $m$.

The proof of the Corollary 2.12.11 goes as follows

(1) since we require the smoothness assumption in the extension theorem we first take some resolution $\mu:V\to \mathcal{X}$ with the strict transform of the central fiber $X_0$ being $X$, and assume $X_0$ is also smooth.

(2) Since both $\mu$ and $\mu|_X$ is birational, thus $K_V+X$ is $\mu$-big. Thus we can apply the extension theorem (Theorem A)

(3) Assume $m$ be the integer that $mK_{X_0}$ is Cartier (always exist some $m$ by canonical singularity assumption)

and we get the surjection $$H^0(V,\mathcal{O}_V(m(K_V+X))\to H^0(\mathcal{X},m(K_{\mathcal{X}}+ X_0))\to H^0(X,mK_X)= H^0(X_0,mK_{X_0})\tag{*}$$ (where the last equality is due to $X_0$ has canonical singularity)

(4) Therefore the nowhere vanishing section of $mK_{X_0}$ lift to nowhere vanishing section on $m(K_{\mathcal{X}}+X_0)$ and lift to section on $m(K_V+X)$

(5) Therefore $K_{\mathcal{X}}+X_0$ is $\mathbb{Q}$-Cartier and $K_V+X\ge \mu^*(K_{\mathcal{X}}+X_0)$.

**The question is** why the surjectivity of (*) implies $K_{\mathcal{X}}+X_0$ is $\mathbb{Q}$-Cartier and $K_V+X\ge \mu^*(K_{\mathcal{X}}+X_0)$?