I am learning Homotopy category of chain complexes from wikipedia. There are two places I do not understand：

1.Let $K(A)$ (respectively, $D(A)$) be a homotopy category (respectively, derived category) of chain complexes. There is a canonical functor $K(A) \rightarrow D(A)$ if $A$ is abelian.

My idea is as follows: $D(A)$ is obtained by localizing $K(A)$ at the set of quasi-isomorphisms. The canonical functor $\pi: K(A) \rightarrow D(A)$ maps $f: M \to N$ to $1_{N} \circ f$. Is my understanding correct？

2.The homotopy category $\textrm{Ho}(C)$ of a differential graded category $C$ is defined to have the same objects as $C$, but morphisms are defined by $\textrm{Hom}_{\textrm{Ho}(C)}(X,Y)=H^{0}\textrm{Hom}_{C}(X,Y)$.

About 2, what is the meaning of $\textrm{Hom}_{\textrm{Ho}(C)}(X,Y)=H^{0}\textrm{Hom}_{C}(X,Y)$?

Edited to add: A differential graded category $C$ (dg category for short) is a category whose morphism sets are complexes: \begin{align*} \cdots \to \textrm{Hom}^{-1}(X,Y) \to \textrm{Hom}^{0}(X,Y) \to \textrm{Hom}^{1}(X,Y) \to \cdots \end{align*} $\textrm{Hom}_{C}(X,Y)=\oplus_{n \in \mathbb{Z}} \textrm{Hom}_{n}(X,Y)$.