Is there a different term for the "left-regular representation" of categories?

Question

After I thought of it, I saw the Yoneda lemma, which initially seemed to be the same thing,
but a more careful examination convinced me that the Yoneda lemma is significantly different.


Let morphismsto and morphismsfrom be given by

morphismsto($A$,$\mathcal{C}\hspace{.03 in}$) is the class of all morphisms to $A$ in $\mathcal{C}$
and
morphismsfrom($A$,$\mathcal{C}\hspace{.03 in}$) is the class of all morphisms from $A$ in $\mathcal{C}$

for categories $\mathcal{C}$ and objects A of $\mathcal{C}$.


For any category $\mathcal{C}$, one can form the "class-category" $\operatorname{L}\hspace{-0.03 in}\operatorname{R}(\mathcal{C}\hspace{.02 in})$ whose objects are
$\{\hspace{-0.03 in}$morphismsto($A$,$\mathcal{C}\hspace{.03 in}) : A\in \mathcal{C}\} \:$ and whose morphisms from $\: \{\hspace{-0.03 in}$morphismsto($A$,$\mathcal{C}\hspace{.03 in}) : A\in \mathcal{C}\} \:$ to
$\{\hspace{-0.03 in}$morphismsto($B$,$\mathcal{C}\hspace{.03 in}) : B\in \mathcal{C}\} \:$ are the class-functions $\;\;\; g \: \mapsto \: f\hspace{-0.05 in}\circ \hspace{-0.04 in}g \;\;\;$ for morphisms $\hspace{.04 in}f$ from $A$ to $B$.
Unless I'm missing something here, one can then define a faithful functor $\mathcal{F}\hspace{.02 in}$ from $\mathcal{C}$ to $\operatorname{L}\hspace{-0.03 in}\operatorname{R}(\mathcal{C}\hspace{.02 in})$
by $\;\;\; \mathcal{F}\hspace{.02 in}(A) \: = \: \{\hspace{-0.03 in}$morphismsto($A$,$\mathcal{C}\hspace{.03 in}) : A\in \mathcal{C}\} \;\;\;$ and $\;\;\; (\mathcal{F}\hspace{.02 in}(\hspace{.05 in}f : A\to B))(\hspace{.02 in}g) \: = \: f\hspace{-0.05 in}\circ \hspace{-0.04 in}g \;\;\;$, $\;\;\;$ and that
$\mathcal{F}\hspace{.02 in}$ is such that for all morphisms $\hspace{.04 in}f\hspace{-0.03 in}$ in $\mathcal{C}$, $\hspace{.04 in}f$ is a monomorphism if and only if $\hspace{.02 in}\mathcal{F}\hspace{.02 in}(\hspace{.05 in}f\hspace{.03 in})$ is injective.

By analogy with groups and rings and algebras, I was imagining that construction would
be called the left-regular representation. $\:$ However, searching with google does not
turn up any use of the phrase "left-regular" in any context like what I'm talking about.

I am well aware that $\operatorname{L}\hspace{-0.03 in}\operatorname{R}(\mathcal{C}\hspace{.02 in})$ can have objects and morphisms which are proper classes
even if $\mathcal{C}$ is locally small. $\:$ Are there any other problems with my (attempted?) construction?
Does my (attempted?) construction have a name?

If my construction works, then in cases where the classes morphismsto($A$,$\mathcal{C}\hspace{.03 in}$) are not necessarily sets but the classes morphismsfrom($A$,$\mathcal{C}\hspace{.03 in}$) are necessarily sets, one can get set objects by applying the construction to the opposite category and then using this answer, although I haven't worked out whether or not that would also give the "monomorphism if and only if $\hspace{.02 in}\mathcal{F}\hspace{.02 in}(\hspace{.05 in}f\hspace{.03 in})$ is injective" property.

Answer 1

Freyd and Scedrov call this construction (seen as a functor $\mathcal{C} \to \text{Set}$ as in Qiaochu Yuan's answer) the Cayley representation in their book Categories, Allegories. They use it to prove the completeness theorem:

Every universally quantified elementary sentence in the predicates of category theory true for the category of sets is true for all categories.

This construction first appeared in the Appendix of the Eilenberg-MacLane paper General theory of natural equivalences, where it is noted that it is an analogue of the left regular representation.

Answer 2

Okay, so here's a version of this construction that makes sense to me. Let $C$ be a small category. There is a functor $C \to \text{Set}$ which takes an object $c \in C$ to the disjoint union $\coprod_{d \in C} \text{Hom}(d, c)$ and which takes a morphism $f : c \to c'$ to the induced morphism

$$\coprod_{d \in C} \text{Hom}(d, c) \to \coprod_{d \in C} \text{Hom}(d, c').$$

This is the composition of the Yoneda embedding $C \to [C^{op}, \text{Set}]$ with the coproduct functor $[C^{op}, \text{Set}] \to \text{Set}$. The Yoneda lemma implies that it's faithful, from which it follows that every small category is concretizable.

I don't know a name for this functor. One way in which it's worse than the Yoneda embedding is that it's not full.