Understanding the proof of Prop 13.11 of Joy of Cats Theorem. Embeddings of dense subcategories preserve limits.

Part of the Proof. Let $\bf{A}$ be a dense subcategory of $\mathbf{B}$ with embedding $E : \bf{A}\to\bf{B}$, and let $D : \bf{I}\to \bf{A}$ be a diagram with a limit $\mathcal{L} = (L\overset{l_i}{\to}D(i))_{i\in \text{Ob}(\mathbf{I})}$. Then $\mathcal{L}$ is a cone over $E\circ D$. Let $\mathcal{S} = (B\overset{f_i}{\to}D(i))_{i\in \text{Ob}(\mathbf{I})}$ be an arbitrary cone over $E\circ D$. By density there exists a diagram $G : \bf{J}\to \bf{A}$ and a colimit $(G(j)\overset{c_j}{\to} B)_{j\in \text{Ob}(\mathbf{J})}$ of $E \circ G$. For each object $j$ of $\text{Ob}(\bf{J})$, $(G(j)\overset{f_i\circ c_j}{\to} D(i))_{i\in \text{Ob}(\mathbf{I})}$ is a cone over $D$. Hence for each $j \in \text{Ob}(\bf{J})$ there exists a unique morphism $g_j : G(j)\to L$ with $f_i \circ c_j = \ell_i \circ g_j$ for each $i \in \text{Ob}(\bf{I})$....

In this proof of Proposition 13.11 of Joy of Cats, I don't understand the reason for writing, the last line. Of course this will happen if $f_i \circ c_j$ is an $\mathbf{A}$-morphism. But I am not sure how that is true unless $\mathbf{A}$ is a full subcategory of $\mathbf{B}$. Could anyone explain?

By definition a (colimit-)dense subcategory is a full subcategory, see definition 12.10 in the linked file. So since $f_i \circ c_j$ is an arrow between two objects in $\mathbf{A}$, that arrow must also be in $\mathbf{A}$. So indeed, the existence of $g_j$ follows from the universal property of the limit $\mathcal{L}$ in $\mathbf{A}$.
