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Let $\mathcal{C}$ be an additive category. Is there a common name for objects $P \in \mathcal{C}$ with the property that $\hom(P,-) : \mathcal{C} \to \mathsf{Ab}$ is right exact, i.e. preserves all finite colimits?

If $\mathcal{C}$ is abelian, this reduces to the requirement that $\hom(P,-)$ preserves epimorphisms. That is, we have the usual notion of a projective object. But in general, right exactness of $\hom(P,-)$ seems to be a stronger condition.

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Kelly [Basic concepts of enriched category theory, §5.5] makes the following definition:

We shall say that an object $A$ in a cocomplete $\mathcal{V}$-category $\mathcal{C}$ is small-projective if the representable $\mathcal{C} (A, -) : \mathcal{C} \to \mathcal{V}$ preserves all small colimits.

On the nLab, such a thing is called a tiny object. Perhaps a variation of one of these would be appropriate for your notion.

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