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added to equalizer statement and proof that a category has equalizers if it has pullbcks and products
With reference to the last diagram, what is an example for a pullback $S\times_{f,g}S$ that isn’t isomorphic to the equalizer? I think another way of asking this is asking for an example where do the two projections out of this pullback objects differ by more than an isomorphism.
And if the category has a terminal object, are this pullback and the equalizer already isomorphic?
Let $f,g$ both be the unique morphism $2\to 1$ in $Set$. Then $S\times_{f,g} S$ is $2\times 2 = 4$, while the equalizer is $2$.
Why not kill two birds with one stone by putting “finite” in parentheses (in appropriate places) in proposition 3.2? The change of the first doesn’t seem entirely warranted to me.
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