Consider a category, whose set of objects is a poset.
Let $f=(f_0,f_1)$ is an indexed family of objects of this category. (Thus $f$ is a 2-indexed family, we can extend it to any index set in an obvious way.)
Let $\prod f$ is the smallest object $\varphi$ such that $\pi_0$ is a morphism $f_0 \rightarrow \varphi$ and $\pi_1$ is a morphism $f_1 \rightarrow \varphi$.
Can we infer from this that $\prod f$ is a direct product in our category?
If, no, under which additional condition we can infer that $\prod f$ is a direct product in our category?