Does a functor which reflects limits also reflect cones?

Question

Following Borceux's Categorical Algebra Definition 2.9.6:

Let $F: \mathcal{C}\to\mathcal{B}$ be a functor. $F$ reflects limits when, for every functor $G: \mathcal{D}\to\mathcal{A}$ with $\mathcal{D}$ a small category and every cone $(L,(p_D)_{D\in\mathcal{D}})$ on $G$, if $(FL, (Fp_D)_{D\in\mathcal{D}})$ is the limit of $F\circ G$ in $\mathcal{B}$, then $(L,(p_D)_{D\in\mathcal{D}})$ is the limit of $G$ in $\mathcal{A}$.

This roughly says that if I have a cone on $G$ in $\mathcal{A}$, I can assert that it's a limit by showing its image under $F$ is a limit of $F\circ G$.

Is it possible to drop the assumption that the collection $(L,(p_D)_{D\in\mathcal{D}})$ is a cone? That is, if $F$ is a limit-reflecting functor, and I have a collection $(L,(p_D)_{D\in\mathcal{D}})$ whose image under $F$ is the limit of $F\circ G$, is that collection a cone (and thus also a limit)?

It's clearly true if $F$ is a faithful functor, so a counterexample would have to be a non-faithful functor which reflects limits.

Answer 1

Given a morphism $f\colon X\to Y$, the pair $(\mathrm{id}_X,f)$ is a cone over a morphism $g\colon X\to Y$, considered as a diagram with domain the category with two objects and one non-identity morphism $\bullet\to\bullet$ if and only if $f=g$, in which case it is a limiting cone of that diagram.

Consequently, a pair of morphisms $f,g\colon X\to Y$ witness that a functor $F$ is not faithful, i.e. $Ff=Fg$ and $f\ne g$ hold, if and only if $(\mathrm{id}_X,f)$ is not a cone over $g\colon X\to Y$, but its $F$-image $(\mathrm{id}_{FX},Ff)$ is a limiting cone over $Fg\colon FX\to FY$.

Thus the question reduces to finding a non-faithful limit-reflective functor. To that end, a cone over a diagram in a groupoid (a category in which every morphism is invertible) is limiting if and only if the domain of the diagram is connected, or the diagram is non-empty and the identity is the only endomorphism of the vertex, or the groupoid is trivial (i.e. there is a unique morphism between any two objects). It follows that the limit-reflecting functors $F$ between groupoids are those which reflect trivial hom-sets, i.e. for which there is a unique morphism $X\to Y$ if there is a unique morphism $FX\to FY$.

Particularly simple non-faithful limit-reflective functors are those between groups considered as one-object categories corresponding to a group homomorphism whose kernel is neither the trivial subgroup, nor the whole group, e.g. corresponding to homomorphisms $\mathbb Z/4\to\mathbb Z/2$.