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Sorry for this silly question but I can't find a reference.

Let $\mathcal C$ be a preadditive category and $A\in \mathcal C$ an object. Under what conditions (on $\mathcal C$) can one say that the functor $\textrm{Hom}_{\mathcal C}(-,A)$ is left exact? Is it true very often or is it specific to categories like module categories?

In particular, I would like to know if $\textrm{Hom}_{S}(-,X)$ is left exact in the category of $S$-schemes, where $X$ is a fixed $S$-scheme.

If $X$ and $S$ are both affine, say $X=\textrm{Spec}\, A$ and $S=\textrm{Spec}\, B$, then \begin{equation} \textrm{Hom}_S(-,X)\cong \textrm{Hom}_B(A,-), \end{equation} and the latter is left exact because $A$ is a $B$-module. But for arbitrary $S$ and $X$?

Thank you for any help.

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    What is the group structure on $Hom_S(-,X)$? How do you define an exact sequence in the category of $S$-schemes? Note that $Hom_B(A,-)$ is not the group of $B$-module homs, its is the set of $B$-_algebra_ homs. - If $C$ is abelian, then $Hom_{\mathcal C}(A, -)$ (and $Hom_{\mathcal C}(-,A)$) is always left exact, e.g. you can use the [Freyd–Mitchell embedding thm](http://en.wikipedia.org/wiki/Mitchell's_embedding_theorem) and proove it in the category of $R$-modules for a suitable ring $R$.2012-07-05
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    I never know what kind of statement one can or cannot prove by using Mitchell Theorem. But probably in this situation one can use it. So you claim also that Sch/$S$ is not even preadditive, right? One can define the product of $f:Y\to X$ and $f':Y'\to X$ to be $f^\ast Y'$, but then I can see no "inverses". Can we already conclude that $\textrm{Hom}_S(-,X)$ is not left exact?2012-07-05
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    Hom is _always_ left exact, for any category whatsoever. To be precise, $\textrm{Hom}(-, X)$ maps all colimits to the corresponding limits. This is straightforward abstract nonsense and follows from the definition of colimit/limit.2012-07-05
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    Thank you, Zhen Lin. And preserving limits (or turning colimits into limits) implies left exact for any category? Perhaps is it even equivalent?2012-07-06
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    @ZhenLin Please consider converting your comment into an answer, so that this question gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141) to make people aware of it (and attract some upvotes). For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113), [here](http://meta.math.stackexchange.com/q/1148) or [here](http://meta.math.stackexchange.com/a/9868).2013-06-09

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