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On p.371 of "The Joy of Cats", by J.Adamek H.Herrlich and G.E.Strecker:

Proposition 21.32

If a topological category $(\mathbf{A},U)$ is a finally dense full concrete subcategory of $(\mathbf{B},V)$, then $(\mathbf{A},U)$ is concretely reflective in $(\mathbf{B},V)$.

Is a concretely reflective full concrete subcategory necessarily finally dense?

Added: By request I recall the definitions involved. A concrete category $(\mathbf{A},U)$ (over the category $\mathbf{X}$) is a category $\mathbf{A}$ together with a faithful functor $U\colon \mathbf{A}\to\mathbf{X}$. A concrete category $(\mathbf{A},U)$ is a concrete subcategory of $(\mathbf{B},V)$ if $\mathbf{A}$ is a subcategory of $\mathbf{B}$ and $U=V\circ E$ where $E\colon\mathbf{A}\hookrightarrow\mathbf{B}$ is the inclusion. A concrete category $(\mathbf{A},U)$ is topological if every $U$-structured source $(X\to UA_i)_{i\in I}$ has a unique $U$-initial lift $(A\to A_i)_{i\in I}$ (equivalently, every $U$-structured sink has a unique $U$-final lift). A concrete subcategory $(\mathbf{A},U)$ of $(\mathbf{B},V)$ is concretely reflective in $(\mathbf{B},V)$ if for each $\mathbf{B}$-object there is an identity-carried $\mathbf{A}$-reflection arrow. A full concrete subcategory $(\mathbf{A},U)$ is finally dense in $(\mathbf{B},V)$ iff for each object $B\in\mathbf{B}$ there is a $V$-final sink $(f_i\colon A_i\to B)_{i\in I}$ in $\mathbf{B}$ with all $A_i\in\mathbf{A}$ (in particular, every $U$-initial source is $V$-initial). I guess this should cover about everything.

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  • $\begingroup$ The terminology and concepts in The Joy of Cats are not necessarily the most standard. It might be helpful to recall the definitions involved. Tentatively, this sounds like a version of Freyd's adjoint functor theorem (with "finally dense" being some analog of the solution set condition), in which case the reflection of an object in $B$ ought to exhibit final density at $B$. $\endgroup$ – Tim Campion Jan 28 '15 at 3:43

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