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Why is the reflection $R(a)$ of an intial object $a$ also initial object in the reflective subcategory (please explain in some detail)? It is nearly obvious but not quite.

$A \subseteq B$ is reflective in $B$ ifand only if there is a functor $R : B\to B$ with values in the subcategory $A$ and a bijection of sets $A(R b, a) \cong B(b, a)$ natural in $b\in B$ and $a \in A$. A reflection may be described in terms of universal arrows: $A \subseteq B$ is reflective if and only if to each $b\in B$ there is an object $R(b)$ of the subcategory $A$ and an arrow $\eta_b : b\to R(b)$ such that every arrow $g: b\to a \in A$ has the form $g = f\circ \eta_b$ for a unique arrow $f: R(b)\to a$ of $A$.

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  • $\begingroup$ For $c\in A$, $A(Ra,c) \simeq B(a,c) \simeq \{*\}$ $\endgroup$ – Max Jul 1 '18 at 14:01
  • $\begingroup$ P.S. $A$ is required to be a full subcategory. $\endgroup$ – Hurkyl Jul 1 '18 at 14:06
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Let $I : A \to B$ be the inclusion functor, and let $R' : B \to A$ be the obvious functor induced by $R$. A more informative way to describe the natural bijection of sets is as giving an adjunction $R' \dashv I$; i.e. where $R'$ is left adjoint to $I$.

It's a general theorem that left adjoints preserve all colimits that exist in the source category. In particular, initial objects are colimits of empty diagrams.

The theorem is actually a one-liner, using the hom-set formulations of colimits and adjunctions:

$$ \hom_A(R'(\mathrm{colim}_i b_i), a) \cong \hom_B(\mathrm{colim}_i b_i, Ia) \cong \mathrm{lim}_i \hom_B(b_i, Ia) \cong \mathrm{lim}_i \hom_A(R' (b_i), a) $$

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