# Constructing counit in adjoint functor theorem for total categories

The theorem I am referring to is,

Let $$C,$$ $$D$$ be locally small categories. Assume $$C$$ is a total category (i.e. the Yoneda functor $$Y : C \to \operatorname{PreSh}(C)$$ has a left adjoint $$Y^L$$). Let $$F:C \to D$$ be a functor which preserves colimits. Then $$F$$ has a right adjoint.

The candidate right adjoint is $$G := Y^L \circ F^{fra}$$, where $$F^{fra}$$ is the formal right adjoint of $$F$$. It remains to construct the unit and counit of the adjunction and check the triangle identities for adjunction (i.e. leave them as an exercise for the reader). For this one explicitly calculates $$G$$ as,

\begin{align*} G(y)= \operatorname{colim}_{w \in C, z \in Hom_{D}(F(w),y)} w. \end{align*}

For brevity I'll denote the above colimit by $$\operatorname{colim}_{w,z} w$$, so $$G(y)= \operatorname{colim}_{w,z} w$$. It's clear to me how the unit is constructed, however I'm stuck on the construction of the counit. It is argued that

\begin{align*} \forall y \in D\ Hom_D(FG(y),y) &= Hom_D(F( \operatorname{colim}_{w,z} w),y)\\ &=Hom_D( \operatorname{colim}_{w,z} F(w)),y)\\ &=\lim_{w,z} Hom_D ( F(w),y). \end{align*}

It is then claimed that $$\forall w \in C$$ there is a canonical element in $$\lim_{w,z} Hom_D ( F(w),y)$$ assigning each $$z \in Hom_D ( F(w),y)$$ to itself. This is the claim giving me trouble. The best I can come up with is that for fixed $$w$$ there are projection maps $$p_{w,z}: \lim_{w,z} Hom_D ( F(w),y) \to Hom(Fw,y)$$ for each $$z \in Hom(Fw,y)$$. Fixing an element $$a \in \lim_{w,z} Hom_D ( F(w),y)$$ we get a map $$z \mapsto p_{w,z(a)}$$ from $$Hom(Fw,y)$$ to itself. But I don't see why there should be an $$a$$ such that this map is the identity.

I'd be really grateful if anyone could help me out here ( any references to a proof of this would also be great). Thanks!

EDIT: From what I've understood from Idéophage's comment:

By the previous argument,

$$Hom_D(FG(y),y)= lim_{w,z} Hom_D(F(w), y)$$.

Because this limit is in set it can be explicitly calculated as,

$$\begin{equation} Hom_D(FG(y),y) = \{(\alpha_{w,z}) | w \in C, z \in Hom_D(Fw,y) \ \text{and} \ \forall \alpha_{w',z'} \in (\alpha_{w,z}) \text{"satisfy some compatibility relation"}\}. \end{equation}$$

I think the map suggested in the comments $$z \to (\alpha_{w,z})_{w}$$ would require $$lim_{w}$$ instead of $$lim_{w,z}$$ ( as mentioned in the comments).

• I don't really understand what you mean by "$\forall w \in C$ there is a canonical element in $\lim_{w,z} Hom_D ( F(w),y)$ assigning each $z \in Hom_D ( F(w),y)$ to itself." What you need is an element of $\lim_{w,z} \operatorname{Hom}_D(F(w),y)$, meaning a collection of maps $F(w) → y$, one for each pair $(w ∈ C, z ∈ \operatorname{Hom}(F(w),y))$. But then you can just associate $z : F(w) → y$ to the pair $(w,z)$. Jan 5 at 16:02
• This is what the quote I extracted from your post seems to mean but it's strangely formulated... Or maybe it should be $\lim_z \operatorname{Hom}(F(w),y)$ instead of $\lim_{w,z}$? Jan 5 at 16:04
• This is great. Thanks for the suggestion, I'll see if I can make sense of it later. Also, I don't really know what the quote means either ( it's not mine) and infact my post was about understanding this quote/ understanding this step in the proof.
– user
Jan 5 at 16:47
• I've added an edit to my post... I think I've understood what you suggested but am stuck on the issue mentioned in your second comment.
– user
Jan 6 at 21:56
• If you are still stuck, maybe you could provide a link to the source? I don't think there is any problem left, it seems to be just a strangely written sentence. You shouldn't worry too much if you understand the idea. Jan 9 at 22:05

You seem to have confused yourself by abbreviating the variables of the limit. What's happening is that the variable $$w\in C$$ specifies that the morphisms in the diagram come from morphisms $$w_1\to w_2\in C$$, while the variable $$z\in Hom_D(F(w),y)$$ indicates that each object has as many copies as elements of $$Hom_D(F(w),y)$$, and that these are labeled by those elements, and which have a morphism from one to the other coming from a morphism $$w_1\to w_2\in C$$ if one is in the image of the other under the corresonding pre-composition function $$Hom_D(F(w_2),y)\to Hom_D(F(w_1),y)$$.

Thus, an element of $$\lim_{w\in C,z\in Hom_D(F(w),y)} Hom_D ( F(w),y)$$ is a family of elements of $$Hom_D((F(w),y)$$, one for each element $$z\in Hom_D(F(w), y)$$, that are compatible in the sense that for any $$g\colon w_1\to w_2$$, the pre-composition function $$Hom_D(Fg,y)\colon Hom_D((F(w_2),y)\to Hom_D((F(w_1),y)$$ sends the element associated to $$z\in Hom_D(F(w_1),y)$$ to the element associated to $$Hom_D(Fg,y)(z)$$. Tautologically, associating to each $$z\in Hom_D(F(w),y)$$ the element of $$Hom_D(F(w),y)$$ that is itself, gives a family satifsying the condition, and thus a (tautological) element of $$\lim_{w\in C,z\in Hom_D(F(w),y)} Hom_D ( F(w),y)$$.

By the way, what you're actually proving is a part of the fact that the above colimit formula (or rather, its post-composition with the colimit-preserving F) produces a left Kan extension of $$F$$ along itself. Left Kan extensions produced by such colimit formulas are by definition the pointwise left Kan extensions, so that's the keywords to look up for more information about the argument you're doing.

• Thanks a lot. Maybe I'm missing the point, but in your post $w$ runs over $C$, right? So shouldn't an element of $\lim_{w\in C,z\in Hom_D(F(w),y)} Hom_D ( F(w),y)$ be a family $(\alpha_{(w,z)})_{w \in C, z \in Hom_D(Fw,y)}$ such that for all $g: w_1 \to w_2$, $Hom(Fg,y)(\alpha_{(w_2,z)})=(\alpha_{(w_1,z \circ Fg)})$? Then letting $\alpha_{(w,z)}=z$ still gives a tautological element, as in your post.
– user
Jan 13 at 9:47
• In the above $z \in Hom(Fw_2,y)$, so $(z \circ Fg) \in Hom(Fw_1, y)$.
– user
Jan 13 at 10:17
• @user yes, $w$ runs over $C$; what I'm getting at is the extension of the indexing notation for limits to indexing notation for weighted limits. Traditionally indices for limits run over categories, but $Hom_D(F(w),y)$ is not a category but the value of a functor to set from the category $C$ (called the weight). This means that each $w\in C$ does not necessarily occur in the diagram. Rather, as many copies of $w$ as elements of $Hom_D(F(w),y)$ do, and the copies are labeled by those elements. Jan 13 at 15:24
• Great! You've cleared it all up for me, thanks!
– user
Jan 13 at 16:11