# Every $\lambda-$pure morphism in a locally $\lambda-$presentable category is a regular monomorphism

Consider the page from the book by Adamek & Rosicky: Locally presentable and accessible categories. given below. I need to derive this: Here is the statement

(2) to prove the universal property of $$f$$, it is sufficient to show that for each $$\lambda-$$presentable object $$H$$, each morphism $$h:H\to B$$ merging $$p$$ and $$q$$ factorizes uniquely through $$f$$.We know that $$f$$ factorizes through some $$b_i$$, say $$h=b_i\cdot h'$$. Then $$d_i^*\cdot (p_i\cdot h')=p\cdot b_i\cdot h'.$$

My question is why this equation holds? All the relevant letters are given here:

And here is the ERRATA to that page from the book.

The equation follows from

$$d^*_i \cdot p_i = p \cdot b_i$$

and this equation holds because of the definition of $$p\colon B\to B^*$$.

Recall that two diagrams of shape $$I$$ are introduced on page 88:

• $$D\colon I\to {\cal K}$$ with $$D_i=B_i$$ and $$D(i,j)=b_{i,j}$$

• $$D^*\colon I\to {\cal K}$$

These have colimits $$B=\mathrm{colim} B_i$$ and $$B^*=\mathrm{colim} D^*_i$$, together with maps $$b_i\colon B_i\to B$$ and $$d^*_i\colon D^*_i \to B^*$$.

The family of maps $$p_i\colon B_i\to D^*_i$$ forms a natural transformation from $$D$$ to $$D^*$$. Therefore the maps $$d^*_i \cdot p_i\colon B_i \to B^*$$ form a cocone which factors through the colimit cocone via a unique map $$p: B\to B^*$$. Similar for the $$q_i\colon B_i\to D^*_i$$ and $$q$$.

• Thank you for your detailed and clear answer. I just do not follow where (and if) we have used the uniqueness of such a $p$? – user122424 Feb 14 '19 at 17:09
• As far as I can see, uniqueness of such a $p$ was not used so far. – Marc Olschok Feb 14 '19 at 18:35