The Yoneda lemma and fully faithful functors

Assume $\mathcal C$ is small and $P:\mathcal C^{\mathrm{op}}\to\mathbf{Set}$ is a presheaf. Put $h_A=\mathcal C(-,A)$. Yoneda gives a bijection between natural transformations $h_A\Rightarrow P$ and elements of $P(A)$. It sends a transformation to its component at $A$ evaluated on the identity arrow. The smallness assumption ensures that the functor category has set-sized Hom collections without further universe bookkeeping; the lemma itself also admits a locally small formulation.

Given $a\in P(A)$, reconstruct the component at $X$ by sending $f:X\to A$ to $P(f)(a)$. The direction is correct because $P(f):P(A)\to P(X)$ reverses the arrow. Naturality of any proposed transformation applied to the arrow $f$ forces exactly this formula. Hence no second transformation can have the same value on the identity, and every choice of that value produces a transformation once the naturality check is made.

For an arrow $g:Y\to X$, the presheaf law gives $P(g)P(f)=P(f\circ g)$. Applying this identity to $a$ proves that the reconstruction commutes with precomposition of probe arrows. Evaluation and reconstruction are inverses because identities act trivially and because the original naturality square determines each component. This is a proof of the bijection, rather than an inference from a coincidental count in finite examples.

The bijection is also natural in $P$ and in $A$. A natural transformation $\beta:P\Rightarrow Q$ carries the chosen element to $\beta_A(a)$, matching composition with $\beta$. An arrow $u:A\to B$ induces postcomposition $h_A\Rightarrow h_B$; precomposing a transformation out of $h_B$ then corresponds to applying $P(u):P(B)\to P(A)$. These directions explain why changing the representing object can reverse arrows in a Hom expression.

The assignment $A\mapsto h_A$ extends covariantly to a functor $y:\mathcal C\to[\mathcal C^{\mathrm{op}},\mathbf{Set}]$ by postcomposition. Taking $P=h_B$ in Yoneda identifies natural transformations $h_A\Rightarrow h_B$ with arrows $A\to B$. Thus $y$ is full and faithful: it gives a bijection on every Hom set. Faithfulness detects equality of arrows, while fullness says every transformation between these represented presheaves comes from an original arrow.

Full faithfulness does not say that every presheaf is represented. It also does not force distinct but isomorphic source objects to become unequal in every possible presentation. Instead it recovers the entire arrow structure between the chosen represented objects. Using covariant representables $\mathcal C(A,-)$ would reverse the direction of the corresponding embedding. Writing the presheaf convention explicitly prevents a false answer even in a tiny ordered example.