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.