Suppose I have $\mathcal{J} = \mathbf{2}$, which is the discrete category with two objects. Now I have a category $\mathcal{C}$, with:
$$ c, c', c''\\ f: c \to c' \\ g: c \to c'' $$
Now I have a functor $D: \mathbf{1} \to \mathcal{C}^\mathcal{J}$, where $\mathcal{C}^\mathcal{J}$ is the category of diagrams. Additionally, I have a functor $\Delta: \mathcal{C} \to \mathcal{C}^\mathcal{J}$. From this, I want to construct a category of cones $\Delta \downarrow D$. In this category there are the following objects and morphisms:
$$ (c,\ast,h_c:\Delta c \to D\ast) \\ (c',\ast,h_{c'}:\Delta c' \to D\ast) \\ (c'',\ast,h_{c''}:\Delta c'' \to D\ast) \\ (f,1_\ast) \\ (g,1_\ast) $$ Now I want to concentrate on $h_c, h_{c'}, h_{c''}$. They all are morphisms in a functor category, which makes them natural transformations, or, as seen in context, cones. However looking back on $\mathcal{C}$, it turns out that not all of these are complete. E.g. $h_{c'}$ could have $1_{c'}$ as one of its natural transformations, but there is no way to get from $c'$ to $c''$ in $\mathcal{C}$, which makes the cone exist in $\mathcal{C}^\mathcal{J}$, but not in $\mathcal{C}$.
Is this okay? Where did I go wrong? Do I have to leave objects which don't form cones out of the category of cones?