Consequences of Monadic Functors Creating Limits

Inclusion of a reflective subcategory creates all limits
A reflective subcategory of a complete category is complete.
Given the p-addic integers defined as the limit of a diagram of shape $ω^{op}$ given as $Z_{p} :\equiv n lim Z / p^{n} \to \dots \to Z / p^{3} \to Z / p^{2} \to Z / p$ can be written as a set (as forgetful functor preserves limits) $Z_{p} = {(a_{1} \in Z / p, a_{2} \in Z / p^{2} \dots) ∣ a_{n} \equiv a_{m} mod p^{min (m, n)}}$ such that the projection maps defined are ring homomorphisms, this tells us that addition and multiplication are componentwise.
$Set$ is cocomplete. As the contravariant powerset functor is monadic
$Mod_{R} \to Ab$ creates all colimits that $Ab$ admits. For any pair of groups $A$ and $A^{'}$ , there is a natural isomorphism $Ab (R \otimes_{Z} A, A^{'}) ≃ Ab (A, Hom_{Z} (R, A^{'}))$ but we know what the tensor product $R \otimes_{Z} -$ has a right adjoint $Hom_{Z} (R, -)$ . Due to LAPC, $R \otimes_{Z} -$ preserves all colimits, so the second point of Monadic Functors Create all limits and some colimits applies to all diagrams.

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