Why is every subobject of a functor a subfunctor?

Question

Let $\mathcal C$ and $\mathcal D$ be categories and let $F:\mathcal C\to\mathcal D$ be a functor. Recall that

• $\theta:G\to F$ is a subobject if it is monic in the functor category $\mathcal D^{\mathcal C}$, and

• $\theta:G\to F$ is a subfunctor if for each $C\in\mathcal C$, the section $\theta_C:G(C)\to F(C)$ is monic.

The nLab article on subfunctors claims that a subfunctor $\theta:G\to F$ of $F$ is the same as a subobject of $F$, and I am having trouble understanding one implication of this claim. It is clear that every subfunctor is a subobject: if the sections of $\theta$ are left cancellative then $\theta$ itself is left cancellative, essentially because equality of natural transformations is checked on the level of sections. But why is the reverse implication true -- why is every subobject of a functor a subfunctor?

My thoughts on this so far: Suppose we know $\theta$ is monic and we want to show that the arrow $\theta_C:G(C)\to F(C)$ is monic. Showing this will require testing $\theta_C$ against arbitrary morphisms $f,g:D\to G(C)$. To me, the natural strategy is to find a functor $P:\mathcal C\to\mathcal D$ and two natural $\tau,\eta:P\to G$ with the following properties:

1. $P(C)=D$
2. $\tau_C=f$ and $\eta_C=g$
3. $\theta\tau=\theta\eta$

An obvious candidate for $P$ is the constant functor $\Delta_D:\mathcal C\to\mathcal D$ (sending every object to $D$ and every morphism to $1_D$), but I can't see how to extend the maps $f$ and $g$ to natural transformations of this functor.

Motivation: This question appears rather technical, but it came up when I was trying to understand subobjects in various categories and got stuck on presheaves. If anyone has wisdom to share about understanding subobjects in sheaf or presheaf categories, I would welcome it.

Answer 1

You're asking if every natural monomorphism is pointwise monic. Since every monomorphism can be characterized as a pullback, this is true if $\mathcal D$ has pullbacks, because pullbacks in $\mathcal D^\mathcal C$ are then computed pointwise, and evaluation functors preserve them.

In general this is not true, but given the weakness of the sufficient condition, you probably won't find many counterexamples, although some have been constructed specifically for this purpose (see this, for example).

However, as a sidenote, you'll find a lot more counterexamples if you're working instead with a subcategory of $\mathcal D^\mathcal C$, especially for the dual claim (epic natural transformation is pointwise epic). This is a famous example.