I've been reading a paper on accessible categories, saturation and categoricity by "Jiří Rosický" for quite some time, but I still cannot understand one detail: In the snippet below, in the Remark 1. why it is sufficient that the inclusion $\mathbf {\text Pure_\lambda}\cal K\to\cal K $ preserves $\lambda^+$-presentable objects ? If someone is interested, I have a yet one more (probably trivial problem) at the very end of the paper which I will add here, in case this question is not going to be closed.
Definition of $\mathbf {\text Pure_\lambda}\cal K$.
EDIT
You also need that $\mathbf{Pure}_\lambda \mathcal{K}$ is $\lambda^+$-accessible, as is mentioned in that remark. Now let $f: K \to L$ be an arrow in $\mathcal{K}$ where $K$ is $\lambda^+$-presentable. Then, as an object in $\mathbf{Pure}_\lambda \mathcal{K}$, we have that $L$ is a $\lambda^+$-directed colimit of $\lambda^+$-presentable objects, so $L = \operatorname{colim}_{i \in I} X_i$ for some $\lambda^+$-directed diagram $(X_i)_{i \in I}$. Here we used that $\mathbf{Pure}_\lambda \mathcal{K}$ is $\lambda^+$-accessible. Note that $\mathbf{Pure}_\lambda \mathcal{K}$ is closed under $\lambda^+$-directed colimits in $\mathcal{K}$, this is always the case (see also point (1) in remark 1 of the paper you ask about). So we have that $L = \operatorname{colim}_{i \in I} X_i$ in $\mathcal{K}$ as well. Since $K$ is $\lambda^+$-presentable and $(X_i)_{i \in I}$ is a $\lambda^+$-directed colimit, we have that $f: K \to L$ factorises as $K \xrightarrow{g} X_i \xrightarrow{h} L$ for some $i \in I$. The arrow $h$ comes from the colimiting cocone and is thus $\lambda$-pure. The object $X_i$ is $\lambda^+$-presentable in $\mathbf{Pure}_\lambda \mathcal{K}$, and by assumption the inclusion preserves $\lambda^+$-presentable objects, so $X_i$ is $\lambda^+$-presentable in $\mathcal{K}$. So we have fulfilled the definition of being weakly $\lambda$-stable, where $X_i$ plays the role of $K'$.