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Fix $A$, and let $S(B)=B⨿A$ be the functor obtained from the chosen binary coproduct in a category construction. Its arrow map is $S(v)=v⨿1_A$. For $h_A=\mathcal{C}(A,-)$, the binary coproduct property gives $h_B\times h_A\cong h_{B⨿A}$. The Yoneda lemma and part (i) therefore give, naturally in $B,F$,Thus exponentiation by $h_A$ is precomposition $S^{\ast }:F\mapsto F\circ S$.

This precomposition has a right adjoint given by Right Kan extension. Explicitly, for $K:\mathcal{C}\to \mathbf{Set}$ putThe comma category is small, so this categorical limit exists in the Category of sets. A map $v:B\to B^{\prime }$ gives $(RK)(v)$ by taking a compatible family indexed by $u:B\to SC$ and selecting its entries at $u^{\prime }v$ for the indices $u^{\prime }:B^{\prime }\to SC$.

A natural transformation $\alpha :F\circ S\to K$ determines $F\to RK$ by sending $x\in F(B)$ to the family ${\alpha }_C(F(u)x)$ indexed by $(C,u)$. Conversely, a map $F\to RK$ determines $F(SC)\to K(C)$ by projecting to the index $(C,1_{SC})$. Naturality and the limit's compatibility relations make these constructions inverse. Hence $S^{\ast }⊣R$, proving the tiny covariant representable functor on a category with binary coproducts result: