= Solution
For the covariant version of the <Yoneda embedding>, put $Y(A)=\mathcal C(A,-)$. A <morphism> $u:A\to B$ induces the <natural transformation> $Y(u):Y(B)\to Y(A)$ whose component sends $v:B\to X$ to $vu$. Thus $Y$ is a <functor> from the <opposite category> $\mathcal C^{\mathrm{op}}$ to the <functor category> $[\mathcal C,\mathbf{Set}]$. The <locally small category> hypothesis makes each value a <set>.
The <Yoneda lemma> says that, for a <functor> $F:\mathcal C\to\mathbf{Set}$, evaluation at the <identity morphism> gives a <natural bijection>
$$
\boxed{\operatorname{Nat}(\mathcal C(A,-),F)\cong F(A),\qquad \theta\longmapsto\theta_A(1_A).}
$$
Given $a\in F(A)$, define $\theta^a_X(v)=F(v)(a)$. For $w:X\to X'$, the <functor> law gives $F(w)\theta^a_X(v)=F(wv)(a)=\theta^a_{X'}(wv)$, proving <naturality>. Conversely, <naturality> of $\theta$ at $v:A\to X$ gives $\theta_X(v)=F(v)\theta_A(1_A)$. These two constructions are inverse, because $F(1_A)(a)=a$.
The <bijection> is natural in both variables. A <natural transformation> $\beta:F\to H$ sends the element $a$ to $\beta_A(a)$, matching postcomposition by $\beta$. For $u:A\to B$, precomposing with $Y(u)$ sends $\theta$ to a transformation from $Y(B)$ and its distinguished element is $\theta_B(u)=F(u)(a)$. This is exactly the required dependence on $A$.
Taking $F=Y(B)$ gives $\operatorname{Nat}(Y(A),Y(B))\cong\mathcal C(B,A)$. Consequently \b[the covariant <Yoneda embedding> is <full and faithful>], with the reversal of arrows accounted for by its <opposite category> domain.
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