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Thm. (AFT) Let $f: \mathsf{A} \to \mathsf{B}$ be a functor preserving colimits from a cocomplete category. The following are equivalent:

 

• For every $b \in \mathsf{B}$, $\mathsf{B}(f\_,b): \mathsf{A}^\circ \to \mathsf{Set}$ is a small presheaf.
• $f$ has a right adjoint.

Thm. (AFT) Let $f: \mathsf{A} \to \mathsf{B}$ be a functor preserving colimits from a cocomplete category. The following are equivalent:

• For every $b \in \mathsf{B}$, $\mathsf{B}(f\_,b): \mathsf{A}^\circ \to \mathsf{Set}$ is a small presheaf.
• $f$ has a right adjoint.

Thm. (AFT) Let $f: \mathsf{A} \to \mathsf{B}$ be a functor preserving colimits between cocomplete categories. The following are equivalent:

Thm. (AFT) Let $f: \mathsf{A} \to \mathsf{B}$ be a functor preserving colimits from a cocomplete category. The following are equivalent: