= Solution
The <Yoneda lemma> states that for $A\in\mathcal C$ and $P:\mathcal C\to\mathbf{Set}$ there is a natural bijection
$$
\operatorname{Nat}(\mathcal C(A,-),P)\cong P(A),
\qquad \alpha\longmapsto\alpha_A(1_A).
$$
If $P\twoheadrightarrow Q$ is an epimorphism in the <functor category>, it is pointwise surjective, so $P(A)\to Q(A)$ is surjective. Yoneda identifies this map with
$$
[\mathcal C,\mathbf{Set}](\mathcal C(A,-),P)
\longrightarrow
[\mathcal C,\mathbf{Set}](\mathcal C(A,-),Q).
$$
Thus 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 functor
$$
[\mathcal C,\mathbf{Set}]\longrightarrow[\mathcal D,\mathbf{Set}].
$$
In 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, then
$$
(P^{yA})(C)
\cong\operatorname{Nat}(yC\times yA,P)
\cong\operatorname{Nat}(y(C\times A),P)
\cong P(C\times A).
$$
Thus 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 gives
$$
[\mathcal C^{\mathrm{op}},\mathbf{Set}](F,P)
\cong(P^F)(1).
$$
Since $F$ 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.
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