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As I understand it, in the category of sets, there is no morphism $\{1\}\rightarrow\emptyset$. On the other hand, is one permitted to say sentences like the following?

"Consider the empty family $(\phi_\alpha)_\alpha$ of morphisms, where $\phi_\alpha:\{1\}\rightarrow\emptyset$."

For example, can one say,

"Find the pullback of the empty family $(\phi_\alpha)_\alpha$ of morphisms, where $\phi_\alpha:\{1\}\rightarrow\emptyset$."

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up vote 4 down vote accepted

Sure. There's still a perfectly well-defined universal property (it's just vacuous) and a perfectly well-defined universal object satisfying it (exercise).

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