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I read, that a comma category $\Delta \downarrow D$ is a category, where the cones are just morphisms. $\Delta$ is the diagonal functor to category of diagrams, $D$ is the diagram.
However, a requirement for a comma category $S \downarrow T$ is that, $S, T$ have a matching codomain, however:
$$ \Delta: \mathcal{C} \to \mathcal{C}^\mathcal{J} \\ D: \mathcal{J} \to \mathcal{C} $$
What am I missing?
Here $D$ is not being thought of as a functor $\mathcal{J}\to\mathcal{C}$. Instead, it is being considered a functor $1\to\mathcal{C}^\mathcal{J}$, where $1$ is the category with one object and one morphism. Such a functor is of course essentially the same thing as an object of $\mathcal{C}^\mathcal{J}$, so any diagram canonically determines such a functor.
