= Exponential of covariant set-valued functors
{title2=$(H^G)(c)=\operatorname{Nat}(\mathcal C(c,-)\times G,H)$}
For a <small category>, this formula defines the <exponential object> in its covariant set-valued <functor category>. An arrow $c\to d$ acts by precomposition with $\mathcal C(d,-)\to\mathcal C(c,-)$. Evaluation applies a natural transformation to $(1_c,z)$; currying a map $P\times G\to H$ sends $p\in P(c)$ to the transformation $(f,z)\mapsto\theta(P(f)p,z)$. The formulas prove the universal property directly, without an adjoint existence theorem.
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