Let $\mathcal{C}$ be a filtered category, $\mathcal{D}$ a finite category and $F:\mathcal{D}\rightarrow\mathcal{C}$ a functor.
The family $(F(D))_{D\in\mathcal{D}}$ is finite; thus, there exists $C\in\mathcal{C}$ and a finite family $(f_D:F(D)\rightarrow C)_{D\in\mathcal{D}}$ of morphisms of $\mathcal{C}$. Note that for each morphism $d:D\rightarrow D'$ of $\mathcal{D}$, there exists $C_d\in\mathcal{C}$ and $g_d:C\rightarrow C_d$ such that $$g_d\circ f_D=g_d\circ(f_{D'}\circ F(d)).$$ Again, we have a finite family $(C_d)_{d\in\text{arr}(\mathcal{D})}$ of objects of $\mathcal{C}$: there exists $C'\in\mathcal{C}$ and a finite family $(h_d:C_d\rightarrow C'_d)_{d\in\text{arr}(\mathcal{D})}$ of morphisms of $\mathcal{D}$. That is to say, we have a finite family $(h_d\circ g_d:C\rightarrow C')_{d\in\text{arr}(\mathcal{D})}$ of morphisms of $\mathcal{C}$. Hence there exists $C''\in\mathcal{C}$ and $k:C'\rightarrow C''$ such that $$k\circ(h_d\circ g_d)=k\circ(h_{d'}\circ g_{d'})$$ for $d,d'\in\text{arr}(\mathcal{D})$.
At this point, the author says,
"Let us write $l$ for this single composite from $C$ to $C''$. The family $(l\circ f_D)_{D\in\mathcal{D}}$ is the required cocone on $F$.
What is $l$? At first, I thought he meant something like $l:=(k\circ(h_d\circ g_d))_{d\in\text{arr}(\mathcal{D})}$. But this doesn't make sense since, for $D\in\mathcal{D}$, the expression "$(k\circ(h_d\circ g_d))_{d\in\text{arr}(\mathcal{D})}\circ f_D$" means nothing. How is the morphism $l:C\rightarrow C''$ defined?