The question and definition at hand are related to the fact that in freshman calculus, to check pointwise continuity of $f:R \rightarrow R$, we ask if both the left and right hand limits exist, and if they agree.
A similar process applies in a category, when we seek to find a product of two objects. The process leads to the notions of left and right side products of two objects in a category. The notions are elementary and likely well studied. The notions are described more or less precisely below, and my question is, what, if any, are the official names of these notions?
Given two objects $x$ and $y$ in a category, and given $p \rightarrow x$, $p \rightarrow y$, to check if $p$ is a product we draw a certain diagram with a missing arrow, and then ask if the missing arrow can be filled in uniquely.
To say that $p$ is a left handed product means , yes we can find a missing arrow, but not necessarily uniquely. However $p$ fails the uniqueness test the most harmlessly.
To say that $p$ is a unique left handed product means if $q \rightarrow x$ and $q\rightarrow y$ has the same properties as $p$, then all the morphisms (built from the defining diagrams) from $p$ to $q$ are isomorphisms.
There is a similar notion of right handed product: A missing arrow might not exist, but if it does it's unique.
Thus if $p$ is both a left and right hand product of $x$ and $y$, then $p$ is a product of $x$ and $y$.
Here are a few more details, to clarify the notion of the left hand product.
Given the data $x$ and $y$, we first consider the collection of all pairs $p \rightarrow x$ and $p \rightarrow y$ so that for all $z$, both $M(z,p) \rightarrow M(z,x)$, and $M(z,p) \rightarrow M(z,y)$ are surjective. This collection admits a nonstrict partial order as follows. Given such a $p$ and such a $q$ in the collection, $q \leq p$ means we have a morphism $ q \rightarrow p$ inducing a surjection $M(z,q) \rightarrow M(z,p)$.
Thus $p$ is a left handed product if $p$ is maximal with respect to the mentioned partial ordering.
