= Solution
Assume the equivalent conditions and that $T$ preserves finite coproducts. The free functor already reflects isomorphisms. By the dual <Crude monadicity theorem>, it remains to preserve the relevant coreflexive equalizers.
A coreflexive equalizer diagram
$$
A\longrightarrow B\rightrightarrows C
$$
with $A\neq\varnothing$ can be equipped with the extra sections making it a split equalizer: choose a point of $A$ and use the common retraction to define the missing splitting maps on the complementary fibres. Every functor preserves split equalizers. If $A=\varnothing$, preservation follows from preservation of the initial object, which follows from preservation of finite coproducts. Consequently $F$ preserves all required coreflexive equalizers, and the adjunction is <comonadic adjunction>[comonadic].
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