# what is this categorical construction?

This is a terminology question.

Consider two morphisms $X \to Y, X \to Z$. Consider all such "wedges" of morphisms $X' \to Y, X' \to Z$ dominated by $X$, i.e. endowed with a morphism $X \to X'$ (and so that the natural diagram commutes). Let $\bar X$ be the "minimal" such wedge, i.e. such that for any wedge $X'$ there is a morphism $X' \to \bar{X}$ (so that the diagrams completed with morphisms to $Y$ and to $Z$ commute).

Is there a common name for $\bar X$?

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Yes: $\bar{X}$ is the product of $Y$ and $Z$ in the category of objects under $X$. If the cartesian product $Y \times Z$ exists in the original category then $\bar{X}$ is that product.
thanks! Though that $\bar X$ is in general $Y \times Z$ cannot be true, I think (unless $X$ is initial) – Dima Sustretov Sep 24 '13 at 18:43