In "Category Theory" by Steve Awodey, there is a suggestion for the reader to construct a left adjoint in the proof of the UMP of the Yoneda embedding. Namely, for any small category $\mathbb{C}$, the Yoneda embedding has the UMP in the sense that for any cocomplete category $\mathcal{E}$ and a functor $F:\mathbb{C} \rightarrow \mathcal{E}$, there is a (unique up to natural isomorphism) colimit preserving functor $F_{!}:\mathbb{Set}^{ \mathbb{C}^{op} } \rightarrow \mathcal{E}$ s. t. $F_{!} \circ y \cong F $ as indicated by the following diagram:

The proof starts with taking some functor $P$ in $\mathbb{Set}^{ \mathbb{C}^{op} }$ and writing it as a colimit of representable functors:

$$ P \cong \lim_{ \rightarrow_{i \in \mathbb{I}} } yC_{i}, $$

where the index category is the category of elements:

$$ \mathbb{I} = \int_{ \mathbb{C} } P.$$

The functor $F_{!}$ acts on objects as follows:

$$ F_{!}(P) = \lim_{ \rightarrow_{i \in \mathbb{I}} } F(C_{i}). $$

Question: how does it act on arrows?

Update 1:

This question

Kan extensions for linear categories

mentions the definition of the Kan extension (an instance of which the Yoneda extension is) taking arrows into consideration as well. It defines the required functor via colimit inclusions, but the method does not look like "direct" or "explicit" construction as it only introduces a condition on arrows. Also, induced inclusions are not completely clear to me, so the definition seems to be circular.

  • $\begingroup$ It's a bit difficult to define directly, but it can be done if you really want to. It's easier to describe the right adjoint and describe the left adjoint in terms of that. $\endgroup$
    – Zhen Lin
    Jan 7, 2015 at 1:05
  • $\begingroup$ I am really interested in explicit constructions, a reference would be appreciated. But the idea of defining the right adjoint first makes sense. Strange that Awodey states the problem the other way around. $\endgroup$
    – Rubi Shnol
    Jan 7, 2015 at 9:35
  • 1
    $\begingroup$ It seems you are neglecting the fact that any natural transformation $P\rightarrow Q$ in $\mathbb{Set}^{ \mathbb{C}^{op} }$ gives a function from elements of $P$ to elements of $Q$. Use that in place of what you call a "family of arrows from $C_i$ to $D_i$. $\endgroup$ Jan 8, 2015 at 17:11
  • $\begingroup$ So, do you see how the function from elements to elements (really, the functor between categories f elements) gives $F_{!}(\vartheta)$? $\endgroup$ Jan 8, 2015 at 19:23
  • $\begingroup$ So, I'm thinking of the following way: consider colimit inclusions object-wise, we are now essentially looking at Hom-sets, we can therefore define the family of functions between yC_i and yD_i. Take y^-1 by: Hom(-,X) -> X. So, we have a family of morphisms f_i:C_i -> D_i and we can define a coproduct arrow F([... ,f_i, f_i+1, ...]) $\endgroup$
    – Rubi Shnol
    Jan 8, 2015 at 20:46

2 Answers 2


The trick is to compare the arrows in $\mathbb{C}$ to those in $\mathbb{Set}^{ \mathbb{C}^{op} }$ or, if you prefer these terms, compare arrows in $\mathbb{C}$ to natural transformations in $\mathbb{Set}^{ \mathbb{C}^{op} }$. You can express this using homsets in the style of your answer but I will express it by diagrams. It would be worth working through explicitly for some small finite categories $\mathbb{C}$. I recommend actually drawing the diagrams.

You have $h:P\rightarrow Q$ in $\mathbb{Set}^{ \mathbb{C}^{op} }$, with $P$ and $Q$ colimits for specified diagrams in $\mathbb{Set}^{ \mathbb{C}^{op} }$. Those are Yoneda images of diagrams in $\mathbb{C}$ so you know how to map those diagrams into $\mathcal{E}$ via $F:\mathbb{C} \rightarrow \mathcal{E}$. Then, as you do in your post you define $F_!$ as mapping colimits to colimits to define $F_!(P)$.

Now the desired $F_!(h)$ is an $\mathcal{E}$ arrow from the colimit of one diagram, to the colimit of another, which by the colimit property means a cone in $\mathcal{E}$ from the first diagram to the colimit of the second.

A vertex of the first cone corresponds to an arrow $k:y(C_i)\rightarrow P$, so $hk:y(C_i)\rightarrow Q$ corresponds to a vertex of the diagram over $F_!(Q)$ and it has a colimit injection (not necessarily monic) to $F_!(Q)$. Simple verification shows these injections commute with the diagram arrows over $F_!(Q)$ and so form a cone from the diagram over $F_!(P)$ to $F_!(Q)$. So they induce an arrow $F_!(h):F_!(P)\rightarrow F_!(Q)$. Trivially, composing this arrow with the one induced by a further $\mathbb{Set}^{ \mathbb{C}^{op} }$ arrow $j:Q\rightarrow R$ is just the same as inducing an arrow by $jh:P\rightarrow R$. It is functorial $\mathbb{Set}^{ \mathbb{C}^{op} } \rightarrow \mathcal{E}$.

  • $\begingroup$ I was thinking in exactly the same way before coming to Hom-sets. The answer is right, but I would be glad to write down a formula in terms of F and h for F_!(h) and I still don't know how to do it. Moreover, I don't know what precisely the correspondence between cone vertices (more generally, arrows) of presheafs and arrows in C (and consequently in E) is, but it's for sure not a functor. Yes, we can map y(C)->(D) to C->D, but there are not only such objects in the presheafs. $\endgroup$
    – Rubi Shnol
    Jan 15, 2015 at 19:02
  • $\begingroup$ But this is the point! You are right there are not only arrows $y(C)\rightarrow y(D)$ among the presheaves. But there are only such arrows in the diagrams used to make each presheaf a colimit of representables. $\endgroup$ Jan 15, 2015 at 19:19
  • $\begingroup$ Ok, my problem was likely that I didn't believe it was so simple. Also, what you call "vertices" of cones I'd rather call "edges" for some reason. I think it would be perfect if we explicitly defined F_! on colimit edges so the mentioned correspondence would be explicitly constructed and we'd be done. $\endgroup$
    – Rubi Shnol
    Jan 15, 2015 at 20:49
  • $\begingroup$ Vertices of a diagram correspond to edges of a colimit cone for the diagram. $\endgroup$ Jan 15, 2015 at 20:59
  • $\begingroup$ Well, of course F_! maps the colimit "injection" arrow inc( y(C_i) ): y(C_i) - > colim(i) y(C_i) to the colimit "injection" arrow inc( F(C_i) ) : F(C_i) - > colim(i) F(C_i). We can finish here. And yes, you are right about the notions. Edges of the colimit are indeed vertices of a diagram. $\endgroup$
    – Rubi Shnol
    Jan 15, 2015 at 21:04

It seems to me the answer is quite simple. On objects $P: C^{op} \to Set$ we have $F_!(P) = \int^{c \in C} P(c) \cdot F(c)$, where for a set $X$, the notation $X \cdot c$ indicates a coproduct of an $X$-indexed collection of copies of $c$. Let $\theta: P \to Q$ be a natural transformation. Then

$$F_!(\theta) := \int^c P(c) \cdot F(c) \stackrel{\int^c \theta(c) \cdot F(c)}{\to} \int^c Q(c) \cdot F(c).$$

Let me write it a little differently, without using coend notation. We have a coequalizer diagram

$$\sum_{c, c'} P(c') \cdot \hom(c, c') \cdot F(c) \rightrightarrows \sum_c P(c) \cdot F(c) \to F_!(P)$$

where one of the parallel arrows uses the covariant action of $C$ on $F$ and the other the contravariant action of $C$ on $P$. Thus, using the universal property of the coequalizer $F_!(P)$, the arrow $F_!(\theta): F_!(P) \to F_!(Q)$ is the unique one making the following diagram commute:

$$\begin{array}{ccccc} \sum_{c, c'} P(c') \cdot \hom(c, c') \cdot F(c) & \rightrightarrows & \sum_c P(c) \cdot F(c) & \to & F_!(P) \\ ^{\sum_{c, c'} \theta(c') \cdot id \cdot id} \downarrow & & ^{\sum_c \theta(c) \cdot id} \downarrow & & \downarrow \\ \sum_{c, c'} Q(c') \cdot \hom(c, c') \cdot F(c) & \rightrightarrows & \sum_c Q(c) \cdot F(c) & \to & F_!(Q) \end{array} $$

where we use the fact that the left and center vertical arrows make the two squares on the left commute serially (this uses naturality of $\theta$ for one of those squares).

  • $\begingroup$ Could you give me a reference to your second equation? I don't completely understand how to interpret the dot in the sum(c) P(c).F(c) $\endgroup$
    – Rubi Shnol
    Jan 15, 2015 at 19:38
  • $\begingroup$ It's also called a copower. See tac.mta.ca/tac/reprints/articles/10/tr10.pdf, page 54 of 143 (page 48 of the book), where the dot notation is used (it's commonly used in category theory). I'm not sure which you meant by the second equation, but essentially I'm using the co-Yoneda lemma: ncatlab.org/nlab/show/co-Yoneda+lemma . The advantage of this formulation (over a category of elements description) is that it generalizes straightforwardly to the enriched context, whereas the category of elements description does not. $\endgroup$
    – user43208
    Jan 15, 2015 at 19:52
  • $\begingroup$ Is theta(c) actually theta indexed by c? Well, I see the point, but Awodey comes to that before discussing copower and coend notions. That's why I don't completely understand that coequalizer and, in particular, the arrow sum(c) theta(c). $\endgroup$
    – Rubi Shnol
    Jan 15, 2015 at 20:24
  • $\begingroup$ Yeah, $\theta(c)$ means the same as $\theta_c$ -- one sees both notations in the literature. The arrow you're asking about, if we rewrite the domain as $\sum_c \sum_{x \in P(c)} F(c)$, is the one whose restriction to the $x^{th}$ copy of $F(c)$ is given by the inclusion of the copy of $F(c)$ indexed by $y = \theta_c(x) \in Q(c)$ in $\sum_c \sum_{y \in Q(c)} F(c)$. Hope that helps. $\endgroup$
    – user43208
    Jan 15, 2015 at 20:38

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