A Topos Has Pullbacks

 Property

Let $\mathcal{E}$ be a topos.  Then $\mathcal{E}$ has pullbacks.  In other words,

 

 

English: For all $f:X \to Z, g:Y\to Z$ in $\mathcal{E}$ there exists maps $p:P \to X, q : P\to Y$ in $\mathcal{E}$ such that the square forms a pullback diagram.

Proof

By definition of topos, namely closure under finite limits, and since a pullback is an example of a finite limit, we immediately get that $\mathcal{E}$ has pullbacks.

$\blacksquare$

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