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The Yoneda lemma states that for $A\in \mathcal{C}$ and $P:\mathcal{C}\to \mathbf{Set}$ there is a natural bijectionIf $P\twoheadrightarrow Q$ is an epimorphism in the functor category, it is pointwise surjective, so $P(A)\to Q(A)$ is surjective. Yoneda identifies this map withThus every representable functor is a projective object in a category.

The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family into a locally small category has a right adjoint functor. For small $\mathcal{C}$, the category $[\mathcal{C},\mathbf{Set}]$ is locally small and has colimits pointwise. Quotients of $P$ are represented by compatible equivalence relations on the sets $P(C)$, so they form a set; hence the category is well-copowered. The set of representables $\{\mathcal{C}(C,-):C\in \mathcal{C}\}$ generates it by the Yoneda lemma. The theorem therefore gives a right adjoint to every small-colimit-preserving functorIn particular, product with a fixed functor $P$ is computed pointwise, and $-\times P(C)$ preserves colimits in the Category of sets. Hence $-\times P$ preserves all small colimits and has a right adjoint $(-{)}^P$. Thus $[\mathcal{C},\mathbf{Set}]$ is a cartesian closed category.

Now work in $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ and write $yA=\mathcal{C}(-,A)$. If $\mathcal{C}$ has binary products, thenThus exponentiation by $yA$ is precomposition with $-\times A$. Precomposition between functor categories has a right adjoint given by Right Kan extension, so $yA$ is a tiny object.

Conversely, suppose $\mathcal{C}$ has a terminal object $1$ and $F$ is tiny. The representable $y1$ is the terminal presheaf, and the exponential adjunction plus Yoneda givesSince $F$ is tiny, $(-{)}^F$ is a left adjoint and preserves all colimits; evaluation at $1$ also preserves pointwise colimits. Therefore the hom functor $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}](F,-)$ preserves coproducts and epimorphisms. Preservation of epimorphisms makes $F$ projective, while preservation of coproducts makes it indecomposable.

Solved by gpt-5.6-sol high.

The comma category $(B\downarrow F)$ has objects $(A,u:B\to FA)$ and morphisms $h:(A,u)\to (A^{\prime },u^{\prime })$ satisfying $F(h)u=u^{\prime }$. An initial object $(LB,{\eta }_B)$ is precisely a universal arrow from $B$ to $F$. Such choices for every $B$ define a functor $L$ and natural bijections $\mathcal{C}(LB,A)\cong \mathcal{D}(B,FA)$, hence a left adjoint functor. The unit of an existing adjunction supplies the initial objects in the reverse direction.

Suppose $F:\mathcal{C}\to \mathcal{D}$ is final and ${\alpha }_A:D(FA)\to X$ is a cocone. For $B\in \mathcal{D}$, choose $(A,u:B\to FA)$ in $(B\downarrow F)$ and defineConnectedness of the comma category makes this independent of the choice, and applying the same argument to arrows proves naturality. Any extension must have this value, so it is unique. Cocones under $D$ and $DF$ are therefore naturally the same; a colimit of the latter is a colimit of the former. Thus existence of all $\mathcal{C}$-shaped colimits in the target implies existence of the required $\mathcal{D}$-shaped colimits.

For arbitrary $F$, define $\mathcal{E}$ to have objects $(B,K)$, where $K$ is a connected component of $(B\downarrow F)$. An arrow $(B,K)\to (B^{\prime },K^{\prime })$ is an arrow $v:B\to B^{\prime }$ whose precomposition functor sends $K^{\prime }$ into $K$. Let $H(B,K)=B$, and let $G(A)=(FA,K_A)$ where $K_A$ contains $(A,1_{FA})$. Then $F=HG$. Given $v:B\to H(B^{\prime },K^{\prime })$, precomposition selects one and only one component $K$, yielding the unique lift $(B,K)\to (B^{\prime },K^{\prime })$; hence $H$ is a discrete fibration. Moreover $((B,K)\downarrow G)$ identifies with the connected component $K$, so it is nonempty and connected. Thus $G$ is a final functor.

Solved by gpt-5.6-sol high.