Lemma 6.15.3. Let (\mathcal{C}, F) be a type of algebraic structure.
\mathcal{C} has a final object 0 and F(0) = \{ * \} .
\mathcal{C} has products and F(\prod A_ i) = \prod F(A_ i).
\mathcal{C} has fibre products and F(A \times _ B C) = F(A)\times _{F(B)}F(C).
\mathcal{C} has equalizers, and if E \to A is the equalizer of a, b : A \to B, then F(E) \to F(A) is the equalizer of F(a), F(b) : F(A) \to F(B).
A \to B is a monomorphism if and only if F(A) \to F(B) is injective.
if F(a) : F(A) \to F(B) is surjective, then a is an epimorphism.
given A_1 \to A_2 \to A_3 \to \ldots , then \mathop{\mathrm{colim}}\nolimits A_ i exists and F(\mathop{\mathrm{colim}}\nolimits A_ i) = \mathop{\mathrm{colim}}\nolimits F(A_ i), and more generally for any filtered colimit.
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