Note: The following construction seems very straightforward, so a pointer to a reference discussing it in detail, or identifying a standard term used to describe the construction which would allow one to find multiple references discussing it, would suffice for an accepted answer.

Background: Given an arbitrary category $C$, it seems that every object $O$ in it can be identified with a category $\mathfrak{O}$ consisting of a single object (i.e. a monoid) whose morphisms are the endomorphisms $O \to O$ in $C$.

So one can construct the category $\mathfrak{C}$ whose objects are the categories $\mathfrak{O}$ and whose morphisms are functors between the categories $\mathfrak{O}$. Question 1: Is there a standard name for this construction?

In particular, given any morphism $f: O_1 \to O_2$ in $C$, one appears to get an induced functor $F: \mathfrak{O}_1 \to \mathfrak{O}_2$ ($e_1 \mapsto f \circ e_1$) and thus also a morphism in $\mathfrak{C}$. Hence it seems like it is always possible to identify $C$ as a subcategory of $\mathfrak{C}$.

Question 2: Are there any interesting examples where $C$ is a strict subcategory of $\mathfrak{C}$? In particular, any interesting examples of functors $F: \mathfrak{O}_1 \to \mathfrak{O}_2$ which are not induced by a morphism $O_1 \to O_2$ in $C$?

Of course, it would be "natural" to consider $\mathfrak{C}$ not just as a category, but also as a 2-category, i.e. to also consider the natural transformations between functors $F: O_1 \to O_2$, $G: P_1 \to P_2$ that constitute the morphisms of $\mathfrak{C}$.

Given any commuting square $h_1: O_1 \to P_1$, $h_2: O_2 \to P_2$ connecting morphisms $f: O_1 \to O_2$, $g: P_1 \to P_2$ in $C$, this obviously induces a commuting square of functors $H_1: \mathfrak{O}_1 \to \mathfrak{P}_1$, $H_2: \mathfrak{O}_2 \to \mathfrak{P}_2$ that connects the functors $F:\mathfrak{O}_1 \to \mathfrak{O}_2$, $G:\mathfrak{P}_1 \to \mathfrak{P}_2$. My understanding is that any commuting square of functors implies a natural transformation (but not necessarily vice versa), so it seems that the 2-category of commutative squares in $C$ can be identified as a sub-2-category of $\mathfrak{C}$.

Question 3: Are there any interesting examples where the commutative squares of $C$ are a strict sub-2-category of $\mathfrak{C}$? I.e. natural transformations in $\mathfrak{C}$ that are not induced by commutative squares of functors?

Or even just natural transformations in $\mathfrak{C}$ that are commutative squares of functors, but at least one of the functors in question is not induced (by morphisms in $C$) and so the natural transformation is also not induced by a commutative square of $C$?

Related questions:

Name for mapping of $(f_1, f_2)$ to $f_2 \circ f_1$?

Commuting square of functors

Why are natural transformations the "right" transformations between functors

Why are natural transformations the morphisms between functors?

What's the name of a morphism the morphism category of the category of categories?

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    $\begingroup$ A morphism $x\to y$ doesn’t induce any function between the endomorphism monoids of $x$ and $y,$ which you seem to be assuming. $\endgroup$ Aug 5, 2023 at 11:32
  • $\begingroup$ Thank you for pointing this out -- I was making that incorrect assumption. $\endgroup$ Aug 5, 2023 at 11:53

1 Answer 1


The induced functor you talk about is not actually a functor except in the trivial cases. Say $O_1\neq O_2$, $f:O_1\to O_2$. If $g:O_1\to O_1$ then $fg:O_1\to O_2$ is not an endomorphism of $O_2$ so you don't have a valid action on arrows (also, even when $O_1=O_2$ your functors are not, in general, preserving the identities).

  • 1
    $\begingroup$ You're right of course. I guess I'll leave the question as is, instead of trying to delete it, in case that somehow prevents other people in the future from making a similar dumb mistake. $\endgroup$ Aug 5, 2023 at 11:51

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