Is the category of finitary $\mathsf{Set}$ endofunctors cartesian closed?
Using the calculus of ends, I could derive a formula for the exponential (this is presented in the article Higher-Order Containers by Altenkirch et al.): $$G^F(X) = \int_Y (X \to Y) \to F Y \to G Y$$
How can I show this limit exists when $F$ and $G$ are finitary? How can I show this expression gives rise to a finitary endofunctor?
I would handle this by observing that your finitary endofunctor category $[\mathrm{Set},\mathrm{Set}]_{\mathrm{fin}}$ is equivalent to the category of all functors $[\mathrm{Set}_{\mathrm{fin}},\mathrm{Set}]$ from the category of finite sets: sets are the free cocompletion under filtered colimits of finite sets. Now, the latter has a Cartesian closed structure defined by the same formula as yours, as does any presheaf category. Thus $[\mathrm{Set},\mathrm{Set}]_{\mathrm{fin}}$ must admit a Cartesian closed structure transferred from that on $[\mathrm{Set}_{\mathrm{fin}},\mathrm{Set}]$. But your formula necessarily computes the homs in any Cartesian closed structure, so this transferred structure is the one you wanted to verify exists.