In the words of the nLab:
In category theory a limit of a diagram $F : D \to C$ in a category $C$ is an object $\lim F$ of $C$ equipped with morphisms to the objects $F(d)$ for all $d \in D$, such that everything in sight commutes. Moreover, the limit $\lim F$ is the universal object with this property, i.e. the “most optimized solution” to the problem of finding such an object.
One obvious way to weaken this is the notion of a weak limit:
A weak limit for a diagram in a category is a cone over that diagram which satisfies the existence property of a limit but not necessarily the uniqueness.
I have encountered a situation where I have the weakening in the opposite direction: I have uniqueness, but not existence. That is, instead of guaranteeing at least one map making a diagram commute, I guarantee at most one. Does this concept, weaker than a limit but not a weak limit, have a name?