$\lambda$-accessible categories, unclear proof In the context of $\lambda$-accessible categories
consider the proof of the proposition $1.22$ here.
How/where did we use the fact that $\cal D$ in the last but one line has less than $\lambda$ morphisms? The proof of proposition $1.16$ is here.
Both proofs are from the book Jiří Adámek, Jiří Rosický: Locally Presentable and Accessible Categories.
 A: The first paragraph of the proof of 1.22 is showing that for any locally $\lambda$-presentable category $\mathcal{K}$ and any object $K$ of $\mathcal{K}$, the canonical diagram with respect to $\mathbf{Pres}_{\lambda}\mathcal{K}$ is $\lambda$-filtered.  In other words, it is showing that if $\mathcal{D}$ is a subcategory of $\mathbf{Pres}_{\lambda}\mathcal{K} \downarrow K$ and $\mathcal{D}$ has less than $\lambda$ morphisms, then the inclusion functor $i : \mathcal{D} \to \mathbf{Pres}_{\lambda}\mathcal{K} \downarrow K$ has a cocone.  
We construct a cocone for $i$ as follows. An object of $\mathbf{Pres}_{\lambda}\mathcal{K}\downarrow K$ is a pair $(X, f)$, where $X$ is a $\lambda$-presentable object of $\mathcal{K}$ and $f : X \to K$ is an arrow in $\mathcal{K}$.  An arrow $g : (X, f) \to (X', f')$ in $\mathbf{Pres}_{\lambda}\mathcal{K} \downarrow K$ is an arrow $g : X \to X'$ in $\mathcal{K}$ such that $f' \cdot g = f$.  There is a natural projection $p : \mathbf{Pres}_{\lambda}\mathcal{K}\downarrow K \to \mathcal{K}$ that sends every pair $(X, f)$ to the object $p(X, f) = X$ of $\mathcal{K}$ and sends every arrow $g : (X, f) \to (X', f')$ to the arrow $p(g) = g : X \to X'$ in $\mathcal{K}$.  The composition $p \cdot i : \mathcal{D} \to \mathbf{Pres}_{\lambda}\mathcal{K}\downarrow K \to \mathcal{K}$ has a colimit, since $\mathcal{K}$ is locally $\lambda$-presentable and thus cocomplete.  Call this colimit $C$.  Unraveling the definition of the colimit, this means that


*

*For every object $D$ of $\mathcal{D}$ there is an arrow $\mathsf{in}_{D} : p(i(D)) \to C$

*For every arrow $g : D \to D'$ in $\mathcal{D}$ we have $\mathsf{in}_{D'} \cdot p(i(g)) = \mathsf{in}_{D}$

*For every object $X$ of $\mathcal{K}$ and every family of arrows $f_{D} : p(i(D)) \to X$ such that $f_{D'} \cdot p(i(g)) = f_{D}$ for every arrow $g : D \to D'$ in $\mathcal{D}$, there is a unique arrow $f : C \to X$ such that $f \cdot \mathsf{in}_{D} = f_{D}$.


We'll use the third bullet point to construct an arrow $C \to K$ in $\mathcal{K}$.  For any object $D$ of $\mathcal{D}$ we have an object $i(D) = (X_{D}, f_{D})$ of $\mathbf{Pres}_{\lambda}\mathcal{K} \downarrow K$, which gives us an arrow $f_{D} : p(i(D)) \to K$.  And for any arrow $g : D \to D'$ in $\mathcal{D}$ we have an arrow $i(g) : i(D) \to i(D')$, which means that we have an arrow $i(g) : X_{D} \to X_{D'}$ such that $f_{D'} \cdot p(i(g)) = f_{D}$.  So by the third bullet point, there is a unique arrow $f : C \to K$ such that $f \cdot \mathsf{in}_{D} = f_{D}$.  This gives us an object $(C, f)$ of $\mathcal{K} \downarrow K$.  And $C$ is a $\lambda$-small colimit of $\lambda$-presentable objects, so $C$ is $\lambda$-presentable by Prop. 1.16.  This means that $(C, f)$ belongs to the subcategory $\mathbf{Pres}_{\lambda}\mathcal{K} \downarrow K$ of $\mathcal{K} \downarrow K$.
Now we show that $i : \mathcal{D} \to \mathbf{Pres}_{\lambda}\downarrow K$ has a cocone with tip $(C, f)$.  For any $D$ in $\mathcal{D}$ we have an object $i(D) = (X_{D}, f_{D})$ of $\mathbf{Pres}_{\lambda}\mathcal{K}\downarrow K$.  We have an arrow $\mathsf{in}_{D} : X_{D} \to C$ in $\mathcal{K}$ by the first bullet point above, and because $f \cdot \mathsf{in}_{D} = f_{D}$ by definition of $f$ we in fact have an arrow $\mathsf{in}_{D} : (X_{D}, f_{D}) \to (C, f)$ in $\mathbf{Pres}_{\lambda}\mathcal{K} \downarrow K$.  For any $g : D \to D'$ in $\mathcal{D}$ we have $\mathsf{in}_{D'} \cdot p(i(g)) = \mathsf{in}_{D}$, and this means that the arrows $\mathsf{in}_{D} : i(D) \to (C, f)$ are a cocone for $i$ with tip $(C, f)$.
So we have shown that any $\lambda$-small subcategory of $\mathbf{Pres}_{\lambda}\mathcal{K}\downarrow K$ has a cocone, which means that $\mathbf{Pres}_{\lambda}\mathcal{K}\downarrow K$ is $\lambda$-filtered.  We used the fact that $\mathcal{D}$ has less than $\lambda$ morphisms when we invoked Prop. 1.16, which tells us that a colimit of a diagram of $\lambda$-presentable objects with less than $\lambda$ morphisms is again a $\lambda$-presentable object.
