= Solution
\b[The Yoneda embedding is an exact representation of a category, but does not by itself reduce the subject to the study of arbitrary functor categories.] There are both a size issue and a structure issue.
Let $\mathcal A$ be a <locally small category> and set $yA=\mathcal A(-,A)$. For a <categorical presheaf> $P:\mathcal A^{\mathrm{op}}\to\mathbf{Set}$, a <natural transformation> $\theta:yA\to P$ is determined by $x=\theta_A(1_A)$. Indeed, naturality along $f:B\to A$ forces
$$
\theta_B(f)=P(f)(x).
$$
Conversely this formula defines a natural transformation for each $x\in P(A)$, since functoriality gives naturality along every $g:C\to B$. Taking $P=yA'$ gives
$$
\boxed{\operatorname{Nat}(yA,yA')\cong\mathcal A(A,A').}
$$
Thus the <Yoneda embedding> is <full and faithful>, and $\mathcal A$ is equivalent to its <full subcategory> of <representable presheaves>. The indexing <category> in the usual version is $\mathcal A^{\mathrm{op}}$, not necessarily a small category chosen independently of $\mathcal A$.
Local smallness makes the values $\mathcal A(B,A)$ <sets>; it does not make the collection of objects a set. For a large indexing category, the collection of <natural transformations> between arbitrary set-valued <functors> can itself be a proper class. For example, take the discrete category on the class of ordinals and the constant two-element functor. Its endomorphisms include an independently chosen function $\{0,1\}\to\{0,1\}$ at each ordinal. A small functor category cannot encode this collection without changing universes. The <Yoneda lemma> remains meaningful when the source is representable because the displayed bijection identifies its natural transformations with a set. It does not make the entire large functor category locally small.
Even for a <small category>, the representable image is usually a very special part of its <presheaf category>. If $\mathcal A$ is the terminal category, its presheaf category is $\mathbf{Set}$, but its representable image consists only of a singleton. The presheaf category has all small <categorical limits> and <colimits>, whereas an arbitrary full subcategory need have almost none. The Yoneda embedding preserves existing limits because <representable functors> preserve limits. It generally does not preserve colimits. For example, the coproduct of two singleton sets in <category of finite sets> is a two-element set, but evaluating the proposed presheaf coproduct at a two-element set gives
$$
|(y1\amalg y1)(2)|=2,\qquad |y(1\amalg1)(2)|=4.
$$
The pointwise coproduct therefore leaves the representable image. Identifying which presheaves are representable, and which ambient constructions return to the original category, is precisely substantive <category theory>.
The representation is nonetheless powerful. The <Yoneda lemma> replaces a morphism by all its probes, converts a <universal property> into a <representation of a functor>, and makes the presheaf category a natural place to add formal <colimits>. A small presheaf category is the free <free cocompletion> under small colimits: every presheaf is a colimit of representables indexed by its <category of elements>. But this is a controlled enlargement of the category, not an identification of its original objects with all functors. The stronger conclusion in the proposed assertion therefore does not follow from the full-faithfulness statement.
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