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can someone help me with the following

Let $C$ be the category with objects subsets of $\mathbb{N}$, and arrows functions $f:A \to B$ such that preimage of each point is a finite set i.e. for every $b \in B$ the set $\{ a \in A \mid f(a) = b \}$ is finite.

Show that $C$ has pullbacks.

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You can express the pullback (in $\mathbf{Set}$) of two maps in terms of the preimages of points of each one. Use this to show that the pullback (in $\mathbf{Set}$) has the required property and thus is in your category. –  Zhen Lin Apr 25 at 9:03
    
I think the pullback of $f$ and $g$ is $\bigcup_{b \in B} f^{-1}(b) \times g^{-1} (b)$, correct? But still it is a product, so not in my category for there we only have subsets of $\mathbb{N}$. –  natural Apr 25 at 10:58
    
Just replace it with another set of the same cardinality. –  Zhen Lin Apr 25 at 12:43

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